Post on 14-Jul-2015
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
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
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
CONTENTS
NATs via overlap integralsTotal state ket of molecules is substituted to the time-dependent Schrödinger eq.:
| (t)i =Z
dRX
n
�n(R, t)|Ri| n;Ri i~ ˙| (t)i = [TN + He]| (t)i
i~Z
dR0X
n0
�n0(R0, t)|R0i| n0 ;R0i = [TN+He(R)]
ZdR0
X
n0
�n0(R0, t)|R0i| n0 ;R0i
6
h n;R|hR|Multiplying from left leads to
i~Z
dR0X
n0
�n0(R0, t)hR|R0ih n;R| n0 ;R0i = i~�n(R, t)�(R�R0) �nn0
Left=
2nd term of right
= h n;R|hR|He(R0)
ZdR0
X
n0
�n0(R0, t)|R0i| n0 ;R0i = ✏n(R)�n(R, t)
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)
Introduction of Nonadiabatic KernelConsidering the infinitesimal time kernel of a molecule
h nf ;Rf |hRf |e�i~ H�t|Rii| ni ;Rii
Trotter decmp.
' h nf ;Rf |hRf |e�i~ TN�te�
ih He(R)�t|Rii| ni ;Rii
9
= h nf ;Rf | ni ;RiihRf |e�i~ TN�t|Riie�
i~ ✏ni (Ri)�t
adiabatic propagation on ni-th adiabatic surface
overlap integral between ni@Ri and nf@Rf
, representing nonadiabatic transition
Repeating this infinitesimal time kernel gives a finite time kernel
= h nf ;Rf |hRf |e�i~ TN�t|Riie�
ih He(Ri)�t| ni ;Rii
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
h n(tk+1);R(tk+1)| n(tk);R(tk)i
Nonadiabatic path integral with overlap integrals
Nonadiabatic Schrödinger eq. with derivative couplings
Mathematically equivalent
14
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
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
Z(�) = Trhe�
�� H · · · e�
�� H
iBoltzmann operator is divided to Γ peaces:
1 =
ZdR
X
n
|Ri| n;Rih n;R|hR|Inserting identity operators
leads to
⇠ =�Y
k=1
h nk ;Rk| nk+1 ;Rk+1i
Z(�) =
ZdR1 · · · dR�
X
n1···n�
⇠hR1|e��� Hn1 |R2i · · · hR�|e�
�� Hn� |R1i
Hn = TN + ✏n(R)
Infinite product of overlaps:
n-th adibatic Hamiltonian:
16
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
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
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
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
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
h n(tk+1);R(tk+1)| n(tk);R(tk)i
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
h n(tk+1);R(tk+1)| n(tk);R(tk)i
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.
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.