20150304 ims mikiya_fujii_dist

Path integral representation and quantum-classical correspondence for nonadiabatic systems

1

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

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

NonAdiabatic Transitions (NATs)

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

◯surface reactions

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

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

◯organic solar cells◯transition probability(’30～)

• Landau–Zener • Stueckelberg • Zhu-Nakamura

◯photo reactions

R. J. Sension et.al, PCCP 16, 4439(2014)

applicationsbasics

◯Theortical methods

T. Kubar and M. Elstner, J. R. Soc. Int. 2013 10, 20130415

Ehrenfest

Surface hopping

http://www.moldyn.uni-freiburg.de/research/ultrafast_nonadiabatic_photoreactions.html

Notations

Electronic HamiltonianHe(R) =

p2

2me+ Vee(r) + VNe(r, R) + VNN (R)

He(R)| n;Ri = ✏n(R)| n;RiTime independent electronic Schrödinger equation

Arbitrary state ket of a molecule

ni-th adiabatic surface

nuclear wavepacket on ni-th adiabatic surface

3

| (t)i =Z

dRX

n

�n(R, t)|Ri| n;Ri

Total Hamiltonian for moleculesH = TN + He(R)

Schrödinger equation for NATsTotal wave function

is substituted to the time-dependent Schrödinger eq.

(r,R, t) =X

n

�n(R, t) n(r;R)

4

i~ (r,R, t) =

� ~22M

@2

@R2� ~2

2me

@2

@r2+ Vee(r) + VNe(r,R) + VNN (R)

� (r,R, t)

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

Nonadiabatic coupling between n-th and n’-th adiabatic surfaces (derivative couplings)

i~�n(R, t) =

�~22M

@2

@R2+ Vn(R)

��n(R, t)�

X

m

~2M

Xnm(R)�0m(R, t) +

~22M

Ynm(R)�m(R, t)

�

Xnm(R) =

Zdr�⇤

n(r;R)@

@R�m(r;R)

Ynm(R) =

Zdr�⇤

n(r;R)@2

@R2�m(r;R)

5


2. Nonadiabatic path integral based on overlap integrals:derivative couplings vs. overlap integrals




CONTENTS

NATs via overlap integrals= h n;R|hR|TN

ZdR0

X

n0

�n0(R0, t)|R0i| n0 ;R0i

=

ZdR0

X

n0

�n0(R0, t)h n;R|hR|TN |R0i| n0 ;R0i

=

ZdR0

X

n0

�n0(R0, t)hR|TN |R0ih n;R| n0 ;R0i

1st term of right

Overlap integral between different nuclear coordinates

i~�n(R, t) =

ZdR0

X

n0

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

Namely,

[

Nonadiabatic interaction between n-th and n’-th adiabatic surfaces

via overlap integrals

7

commutable

Differential form vs. integral form of Schrödinger equation

i~�n(R, t) =

ZdR0

X

n0

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

◯differential form: NATS via derivative couplings

◯Integral form: NATs via overlap integrals

They are Mathematically equivalent

Nonlocal propagation from R’ to R ↓　

Suitable for the path integral representation

8

i~�n(R, t) =

� ~22M

@2

@R2� ✏n(R)

��n(R, t)

�X

n0

~2M

⌧ n(r;R)

��@

@R n0(r;R)

�@

@R�n0(R, t)�

X

n0

~22M

⌧ n(r;R)

��@2

@R2 n0(r;R)

��n0(R, t)

10

K =

ZD [R(⌧), n(⌧)] ⇠ exp

i

~S�

②Infinite product of the overlap integrals

(phase weighted probability of each path)

①Nuclear paths that are evolving through arbitrary

positions and electronic eigenstates{R(⌧), n(⌧)}

NonAdiabatic Path Integral (NAPI)

This nonadiabatic kernel holds 2 differences from adiabatic kernel

⇠ ⌘ lim�!1

�Y

k=0

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

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

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

NA Schrödinger eq. is revisited from the NAPI

�n(x, t+ ✏) =

X

m

Z 1

�1d⌘hn;x|m;x+ ⌘i exp

i

~⌘

2

2✏

� i

~Vm(x+ ⌘)✏

��m(x+ ⌘, t)

Time propagation with infinitesimal time-width in NAPI:✏

�p2~✏ < ⌘ <

p2~✏

The main contribution is from the range: ⌘2

2~✏ ' 1

i.e.,

Then, we expand the NAPI up to .✏ or ⌘2

�n(R, t+ ✏) =

X

m

Z 1

�1d⌘A exp

�M⌘2

2i~✏

�⇢hn;R|m;Ri�m(R, t) +

1

i~ hn;R|m;RiVm(R)�m(R, t)✏

+hn;R|m;Ri@�m

@R⌘ +Xnm(R)�m(R, t)⌘

+hn;R|m;Ri@2�m

@R2

⌘2

2

+Xnm(R)

@�m

@R⌘2 + Ynm(R)�m(R, t)

⌘2

2

�,

�nm

By solving the Gaussian integrals, the nonadiabatic Schrödinger eq. is revisited:

i~�n(R, t) =

�~22M

@2

@R2+ Vn(R)

��n(R, t)�

X

m

~2M

Xnm(R)�0m(R, t) +

~22M

Ynm(R)�m(R, t)

�

Xnm(R) =

Zdr�⇤

n(r;R)@

@R�m(r;R)

Ynm(R) =

Zdr�⇤

n(r;R)@2

@R2�m(r;R)

K =

ZD [R(⌧), n(⌧)] ⇠ exp

i

~S�

i~�n(R, t) =

�~22M

@2

@R2+ Vn(R)

��n(R, t)�

X

m

~2M

Xnm(R)�0m(R, t) +

~22M

Ynm(R)�m(R, t)

�

Xnm(R) =

Zdr�⇤

n(r;R)@

@R�m(r;R)

Ynm(R) =

Zdr�⇤

n(r;R)@2

@R2�m(r;R)

⇠ ⌘ lim�!1

�Y

k=0


Nonadiabatic path integral with overlap integrals

Nonadiabatic Schrödinger eq. with derivative couplings

Mathematically equivalent

14






CONTENTS

Z(�) = Tre��H

Quantum MC by Adiabatic beads

Quantum MC by Nonadiabatic beads

Nonadaibatic Partition function

K = e�i~ Ht

t = �i~�time propagator partition function

17

The divided Boltzmann operators can be written as hR|e�

�� Hn |R0i = lim

�!1⇢0(R,R0;

�

�)e�

�� ✏n(R

0)

⇢0(R,R0;�

�) =

✓M�

2⇡~2�

◆ 12

e� M�

2~2�(R�R0)2

Boltzmann operator for free particles

After all, we obtained following representation:

Z(�) = lim

�!1

✓M�

2⇡~2�

◆�2 X

n1,··· ,n�

ZdR1, · · · , dR�

⇥⇠ exp

� �

✓ �X

k=1

M�

2~2�2(Rk �Rk+1)

2+

✏nk(Rk)

�

◆�

nonadiabatic beads

quantum-classical mapping under thermal equilibrium

Z(�) = Tre��HTo calculate the partition function:

Hbeads =�X

k=1

M�

2~2�2(Rk �Rk+1)

2 +✏nk(Rk)

�

with weighting factor:

⇠ =�Y

k=1

h nk ;Rk| nk+1 ;Rk+1iThis nonadiabatic beads model can be applied to thermal average of physical quantities

“quantum” nonadiabatic particle “classical” nonadiabatic beads

18

classical mapping

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

20

✏ n(R)[kcal/m

ol]

✏ n(R)[kcal/m

ol]

Black: Adiabatic energy levels Red: Nonadiabatic energy levels

Numerical example: thermal average

adiabatic (exact)

nonadiabatic (exact)

nonadiabatic beads

“quantum-classical mapping” under thermal equilibrium

To calculate the partition function and thermal average

Hbeads =�X

k=1

M�

2~2�2(Rk �Rk+1)

2 +✏nk(Rk)

�

with weighting factor:

⇠ =�Y

k=1

h nk ;Rk| nk+1 ;Rk+1i

classical mapping

“quantum” nonadiabatic particle “classical” nonadiabatic beads

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

23






CONTENTS

Semiclassical propagator (adiabatic)Stationary phase approx. is applied to the time propagatorK = hRf |e�iH(tf�ti)/~|Rii =

ZD[R(⌧)] exp

i

~S[R(⌧)]

�

Ri

R1

R2

Rf

RN�1

RN

R0

t1

t2

tN�1

tf

ti

t

stationary phase condition：minimum action integral→classical trajectory: �S[R(⌧)]

�R(⌧)= 0

Rcl(⌧)

Rcl(⌧)

� : Maslov indexS[Rcl(⌧)]: action integraldRt

dPi: Stability matrix

Formulated with quantities along classical trajectories

“Quantum-Classical correspondence” in dynamics

KSC =

X

Rcl

(2⇡i~)� 12

��dRt

dPi

�� 1

2

exp

i

~Scl[Rcl(⌧)]�i⇡

2

⌫

�

24

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

K /Z

dth

ZdRh⇠J exp

i

~SnJcl (Rf , Rh)

�⇠I exp

i

~SnIcl (Rh, R0)

�

25

Semiclassical approximation of the nonadiabatic kernel (stationary phase approximation on the each surface)

K /Z

dth

ZdRh⇠J exp

i

~SnJcl (Rf , Rh)

�⇠I exp

i

~SnIcl (Rh, R0)

�

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

d

dRh[SnJ

cl (Rf , Rh) + SnIcl (Rh, R0)] = �PJ + PI = 0

Stationary phase condition:

momentum conservation26

Semiclassical approximation of the nonadiabatic kernel (stationary phase approximation for the hopping point)

27

Nonadiabatic Semiclassical Kernel

c.f., Adiabatic semiclassical kernel

KSC =

X

Rcl

(2⇡i~)� 12

��dRt

dPi

�� 1

2

exp

i

~Scl[Rcl(⌧)]�i⇡

2

⌫

�

KSC =

X

Rhcl

(2⇡i~)� 12 ⇠

��dRt

dPi

�� 1

2

exp

i

~Scl[Rhcl(⌧)]�i⇡

2

⌫

�

①Hopping classical trajectoriesTwo differences from adiabatic semiclassical kernel

⇠ ⌘ lim�!1

�Y

k=0

h n(tk+1);R(tk+1)| n(tk);R(tk)iamplitude of each overlap means probability of the hopping at each time step

②Infinite product of the overlap integrals

(phase weight probability of each hopping calssical traj.)

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

M. Fujii, J. Chem. Phys. 135, 114102 (2011)28

Black: Numerical exact Blue＆Green: present semi classical 107 trajectories

avoided crossing

Nonadiabatic wavepacket dynamics including phase accompanied by

nonadiabatic transition is also reproduced. Namely, “rigorous” surface hopping.

M. Fujii, J. Chem. Phys. 135, 114102 (2011)29

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

“Quantum-classical correspondence” in nonadiabatic dynamics

“quantum” wavepacket dynamics “classical” hopping dynamics

Classical hopping trajectories are taken out as dominant terms of nonadiabatic propagation of quantum wavepackets

stationary phase

31






CONTENTS

Semiclassical Quantization

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

H| i = E| i

steady states in quantum mechanics

q

p

time-invariant structures in phase space of

classical mechanics

periodic orbits torus

big←　→small~

32

Objective

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))

big←　→small~nonadiabatic eigenstates

q

p

？nuclear phase space

Gutzwiller’s trace formulaSemiclassical approximation to DOS, which has revealed correspondence between quantum energy levels and classical periodic orbits through divergences of DOS.

classical action: Phase space volume

⌫ = 2

Scl = 2⇡E/!

e.g. Harmonic oscillator

Maslov index: number of intersects between trajectory and R-axis

geometric quantity of a cycle of primitive

periodic orbit

number of cycle of primitive periodic orbit

Sum of k-cycle diverges at quantum energy levels

1 = exp

✓i

~2⇡E

!� i⇡

◆�

) En =

✓n+

1

2

◆~!

}⌦(E) /

1X

k=0

exp

✓i

~Scl � i⇡

2

⌫

◆�k=

1� exp

✓i

~Scl � i⇡

2

⌫

◆��1

34

⌦(E) /X

�2PHPO

1� ⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆��1

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

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

⇠� < 1This term does not diverge because .

35

②Infinite product of the overlap integrals: ⇠ ⌘ lim�!1

�Y

k=0


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

⌦(E) /X

�2PHPO

1X

k=0

⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆�kNonadiabatic Trace formula

That is, individual PHPO cannot be quantized.

We must introduce another way to take the summation of infinite number of the PHPOs

Bit sequence which represents PHPO A concrete example: Two adiabatic harmonic oscillators which interact nonadiabatically at the origin only.D12 = �(R) sin(✓)

Ri Rj Rk Rl 0, 1, 1, and 0 are assigned when a trajectory passes through Ri, Rj, Rk, and Rl,

respectively

Assignment of bit

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

0111 ⌘ 011101110111 · · ·01 ⌘ 0101010101 · · ·

36

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 infinite.

D12 = �(R) sin(✓)

Ri Rj Rk Rl 0100011100110 in odd-numbered bits means “returning to Ri”.

Decomposition of each PHPO

01 + 00 + 0111 + 0011

At the 0 in odd-numbered bits, we can decompose this PHPO to “more primitive (prime) bits (PHPOs)”.

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 ⌘

(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

38

S0 ⌘00, 01, 0110, 0111, 0010, 0011

A set ”Si” of prime PHPOs

D12 = �(R) sin(✓)

Ri Rj Rk Rl

Sum of all HPOs as combination of coprime PHPOs

Sum of all HPOs, for example, started from Ri

D12 = �(R) sin(✓)

Ri Rj Rk Rl00

01+

+0110

+

...0000

0101+

+000110...000000

010101...

k = 1

k = 2

k = 3

}}}

= Sum of geometric series of sum of prime PHPOs

=X

k2N(00 + 01 + 0110)k =

X

k2N

X

�2S0

�

!k

Semiclassical(nonadiabatic)Exact(nonadiabatic)Exact(adiabatic)40

S0 ⌘

⌦(E) /X

Si2{S}

1X

k=0

"X

�2Si

⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆#k

=

X

Si2{S}

"1�

X

�2Si

⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆#�1

Divergence points give quantum levels =quantum condition

sum of all prime PHPOs

D12 = �(R) sin(✓)

Ri Rj Rk Rl

!I = 27.6 [kcal

1/2mol

�1/2˚

A

�1amu

�1/2]

!II = 38.64 [kcal

1/2mol

�1/2˚

A

�1amu

�1/2]

m = 1[amu]

“Quantum-classical correspondence” in nonadiabatic steady states

“quantum” nonadiabatic eigenstates

Time-invariant structure in “classical” nuclear phase space

big←　 →small~ S0 ⌘

1 =

X

�2S

⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆Semiclassical Quantization condition

Summary of this talk

S0 ⌘

1. Nonadiabatic path integral with overlap integrals

K =

ZD [R(⌧), n(⌧)] ⇠ exp

i

~S�

⇠ ⌘ lim�!1

�Y

k=0


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

KSC =

X

Rhcl

(2⇡i~)� 12 ⇠

��dRt

dPi

�� 1

2

exp

i

~Scl[Rhcl(⌧)]�i⇡

2

⌫

�

4. Semiclassical quantization condition

⌦(E) /X

�2PHPO

1X

k=0

⇠� exp

✓i

~Scl� � i⇡

2

⌫�

◆�kNonadiabatic trace formula

42

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

M. Fujii and K. Yamashita, JCP, 142, 074104 (2015) arXiv:1406.3769

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

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

2. Nonadiabatic beads

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

Classical mapping under thermal equilibrium

Classical counterparts of nonadiabatic wavepacket dynamics

Classical counterparts of nonadiabatic eigenstates

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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