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)
⃝vision
X.-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

Notations
Electronic Hamiltonian
ˆHe( ˆR) =
ˆp2
2me
+ Vee(ˆr) + VNe(ˆr, ˆR) + VNN ( ˆR)
ˆHe(R)| n; Ri = ✏n(R)| n; Ri
Time 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
dR
X
n
n(R, t)|Ri| n; Ri
Total Hamiltonian for molecules
ˆH = ˆTN + ˆHe( ˆR)

Schrödinger equation for NATs
Total 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) =

~2
2M
@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) =

~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m

~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
n(r; R)
@2
@R2 m(r; R)

5
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
CONTENTS

NATs via overlap integrals
= h n; R|hR| ˆTN
Z
dR0
X
n0
n0 (R0
, t)|R0
i| n0 ; R0
i
=
Z
dR0
X
n0
n0 (R0
, t)h n; R|hR| ˆTN |R0
i| n0 ; R0
i
=
Z
dR0
X
n0
n0 (R0
, t)hR| ˆTN |R0
ih n; R| n0 ; R0
i
1st term
of right
Overlap integral between
diﬀerent nuclear coordinates
i~ ˙n(R, t) =
Z
dR0
X
n0
hR| ˆTN |R0
ih n; R| n0 ; R0
i 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) =
Z
dR0
X
n0
hR| ˆTN |R0
ih n; R| n0 ; R0
i 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
representation8
i~ ˙n(R, t) =

~2
2M
@2
@R2
✏n(R) n(R, t)
X
n0
~2
M
⌧
n(r; R)
@
@R
n0 (r; R)
@
@R
n0 (R, t)
X
n0
~2
2M
⌧
n(r; R)
@2
@R2 n0 (r; R) n0 (R, t)

10
K =
Z
D [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
1
d⌘hn; x|m; x + ⌘i exp

i
~
⌘2
2✏
i
~
Vm(x + ⌘)✏ m(x + ⌘, t)
Time propagation with inﬁnitesimal time-width in NAPI:✏
p
2~✏ < ⌘ <
p
2~✏
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
1
d⌘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) =

~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m

~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
n(r; R)
@2
@R2 m(r; R)

K =
Z
D [R(⌧), n(⌧)] ⇠ exp

i
~
S
i~ ˙n(R, t) =

~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m

~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
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
ˆH
Quantum MC
by Adiabatic beads
Quantum MC
by Nonadiabatic beads
Nonadaibatic Partition function
K = e
i
~
ˆHt
t = i~time propagator partition function

17
The divided Boltzmann operators can be written as
hR|e
ˆHn
|R0
i = lim
!1
⇢0(R, R0
; )e ✏n(R0
)
⇢0(R, R0
; ) =
✓
M
2⇡~2
◆1
2
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
Z
dR1, · · · , 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
ˆHTo 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/mol]
✏n(R)[kcal/mol]
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 propagator
K = hRf |e i ˆH(tf ti)/~
|Rii =
Z
D[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 index
S[Rcl(⌧)]: action integral
dRt
dPi
: Stability matrix
Formulated with quantities along classical trajectories
Quantum-Classical correspondence in dynamics
KSC =
X
Rcl
(2⇡i~)
1
2
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
Z
dRh⇠J exp

i
~
SnJ
cl (Rf , Rh) ⇠I exp

i
~
SnI
cl (Rh, R0)
25
Semiclassical approximation of the nonadiabatic kernel
(stationary phase approximation on the each surface)

K /
Z
dth
Z
dRh⇠J exp

i
~
SnJ
cl (Rf , Rh) ⇠I exp

i
~
SnI
cl (Rh, R0)
(b)→(c): Stationary approximation for the integral related to hopping points, Rh
d
dRh
[SnJ
cl (Rf , Rh) + SnI
cl (Rh, R0)] = PJ + PI = 0
Stationary phase condition:
momentum conservation
26
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~)
1
2
dRt
dPi
1
2
exp

i
~
Scl[Rcl(⌧)]
i⇡
2
⌫
KSC =
X
Rhcl
(2⇡i~)
1
2 ⇠
dRt
dPi
1
2
exp

i
~
Scl[Rhcl(⌧)]
i⇡
2
⌫
①Hopping classical trajectories
Two differences from adiabatic semiclassical kernel
⇠ ⌘ lim
!1
Y
k=0
amplitude 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
ˆH| 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 formula
Semiclassical 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
⌫
◆ k
Nonadiabatic 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
˙011˙1 ⌘ 011101110111 · · ·
˙0˙1 ⌘ 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 ˙01000111001˙1
0 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 Rl
00
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 [kcal1/2
mol 1/2˚A 1
amu 1/2
]
!II = 38.64 [kcal1/2
mol 1/2˚A 1
amu 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 =
Z
D [R(⌧), n(⌧)] ⇠ exp

i
~
S
⇠ ⌘ lim
!1
Y
k=0
3. Nonadiabatic semiclassical kernel (“rigorous” surface hopping)
KSC =
X
Rhcl
(2⇡i~)
1
2 ⇠
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
⌫
◆ k
Nonadiabatic 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.
本研究のみならず，有機薄膜太陽電池やポストリチウムイオ
ン電池など山下・牛山研の全ての研究において分子科学研究
所計算科学研究センターに大変お世話になっております．管
理運営されている先生およびスタッフの皆様に心より御礼申
し上げます．

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