26
Ultrafast processes in molecules Mario Barbatti [email protected] VIII – Methods in quantum dynamics
2
sN
kkk
1
;, RrRRr kk nuclear wave function
Multiply by i at left and integrate in the electronic coordinates
Time dependent Schrödinger equation for the nuclei
sN
kkkNikiiiN T
tiET
ti
1
0
eN HT H
0),,(
tt
i RrH
Quantum dynamics
3
1
1
1.11 1
)(1
)1()(...,
n
j
n
jf
fjfj
ijji
f
f
fRRAt R
Expand nuclear wave function in a basis of nk functions k for each coordinate Rk:
sN
kkkNikiiiN T
tiET
ti
1
0
k
kiki
ti )()( AHΑ
Wavepacket program:http://page.mi.fu-berlin.de/burkhard/WavePacket/
Kosloff, J. Phys. Chem. 92, 2087 (1988)
Wave-packet propagation
4
5
Multiconfiguration time-dependent Hartree (MCTDH) method
1
1
1.11 1
)(1
)1()(... ,,)(,
n
j
n
jf
fjj
ijji
f
f
fftRtRtAt R
Variational method leads to equations of motion for A and .
Heidelberg MCDTH program:http://www.pci.uni-heidelberg.de/tc/usr/mctdh/
Meyer and Worth, Theor. Chem. Acc. 109, 251 (2003)
Conventional: 1 – 4 degrees of freedomMCTDH: 4 -12 degrees of freedom
6
UV absorption spectrum of pyrazine (24 degrees of freedom!)
MCTDH Exp.
Raab, et al. J. Chem. Phys. 110, 936 (1999)
tii etdtI 01
7
Multiple spawning dynamics
tN
m
Nim
im
im
im
tiimi
i atim PRRgetct
1
3
1
,,;;
R
im
im
im
im
imi
mim
im
im RRPiRRPRRg
24/1
exp2
,,;
Nuclear wavefunction is expanded in a Gaussian basis set centered at the classical trajectory PR,
Note that the number of gaussian functions (Ni) depends on time.
Ben-Nun, Quenneville, Martínez, J. Phys. Chem. A 104, 5161 (2000)
8
Multiple spawning dynamics
tN
m
Nim
im
im
im
tiimi
i atim PRRgetct
1
3
1
,,;;
R
sN
kkeikkNikiN HTT
ti
1
0
0NT ijjn
im
ijmn ggS with and
ij
jijiiiiiiii
iidtd CCSSC 1
9
Multiple spawning dynamics
The non-adiabatic coupling within Hij is computed and monitored along the trajectory.
When it become larger than some pre-define threshold, new gaussian functions are created (spawned).
10
Multiple spawning dynamics
jk
JNiMijiNijiij TETH
hH 2
Saddle point approximation
Rff jiji
11
S1 S0
12
QM (FOMO-AM1)/MM
Virshup et al. J. Phys. Chem. B 113, 3280 (2009)
13
sN
kkkNikiiiN T
tiET
ti
1
0
Global nature of the nuclear wavefunction
R’R
E
tt’
E1 ''1 tR
kt
1 ttE '1 R
12 M
14
sN
kkkiii t
iEt
i1
0
Global nature of the nuclear wavefunction
R’R
E
tt’
E1 ''1 tR
kt
1 ttE '1 R
02
2 iMiN MT
15
Within this approximation, the nuclear wavefunction is local: it does not depend on the wavefunction values at other positions of the
space
This opens two possibilities:1) On-the-fly approaches (global PESs are no more needed)
2) Classical independent trajectories approximations
However, because of the non-adiabatic coupling between different electronic states, the problem cannot be reduced to the Newton’s
equations
We use, therefore, Mixed Quantum-Classical Dynamics approaches (MQCD)
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Ehrenfest (Average Field) Dynamics
tHt eNc RF
With
sN
k
ckk ttt
1
;Rr
i which solves: 0 ie EH (adiabatic basis)
Weighted average over all gradients
Meyer and Miller, J. Chem. Phys. 70, 3214 (1979)
s
c
N
kkk
c Et1
2RRRF
Prove!
0 kNRemember we assumed
17
t
E
0)0( 20
1)0( 21
0 tci
5.0)( 21 CIt
5.0)( 20 CIt
8.0)( 20
2.0)( 21
Average surface
18
s*
ns*
cs
Energy
N-H dissociation
10 fs
*
Ehrenfest dynamics fails for dissociation
19
0 2 4 6 8 10 12 14
-224.85
-224.80
-224.75
-224.70
-224.65
s*
cs
Ene
rgy
(au)
Time (fs)
cs
s*
0.43
0.57
In this case, Ehrenfest dynamics would predict the dissociation on a
surface that is ~half/half S0 and
S1, which does not correspond to the
truth.
Landau-Zener predicts the populations at t = ∞
ss
212
** 2expH
PtHH 2211
vF12 1
20
The problem with the Ehrenfest dynamics is the lack of decoherence.
The non-diagonal terms should quickly go to zero because of the coupling among the several degrees of freedom.
The approximation i(R(t)) ~ i(t) does not describe this behavior adequately.
Ad hoc corrections may be imposed.
Zhu, Jasper and Truhlar, J. Chem. Phys. 120, 5543 (2004)
Decoherence
2
10*1
1*0
20
21
Surface hopping dynamics
iNc ERF
Tully, J. Chem. Phys. 93, 1061 (1990)
Dynamics runs always on a single surface (diabatic or adiabatic).
Every time step a stochastic algorithm decides based on the non-adiabatic transition probabilities on which surface the molecule will stay.
The wavepacket information is recovered repeating the procedure for a large number of independent trajectories.
Because the dynamics runs on a single surface, the decoherence problem is largely reduced (but not eliminated).
22
t
E
0)0( 20
1)0( 21
0 tci
5.0)( 21 CIt
5.0)( 20 CIt
8.0)( 20
2.0)( 21
23
Tully, Faraday Discuss. 110, 407 (1998).
Burant and Tully, JCP 112, 6097 (2000)
wave-packet
surface-hopping (adiabatic)mean-field
Landau-Zener
surface-hopping (diabatic)
Comparison between methods
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Worth, hunt and Robb, JPCA 127, 621 (2003).
Oscillation patterns are not necessarily quantum interferences
Butatriene cation
Barbatti, Granucci, Persico, Lischka, CPL 401, 276 (2005).
Ethylene
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1
1
1.11 1
)(1
)1()(... ,,)(,
n
j
n
jf
fjj
ijji
f
f
fftRtRtAt R
Quantum
Classical
Multiple spawning
MQCD dynamics
Wave packet (MCTDH)
tN
m
Nim
im
im
im
tiImi
i atim PRRgetct
1
3
1
,,;;
R
2/1, c
ii tt RRR
Hierarchy of methods
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Next lecture:
• Spectrum simulations• Implementation of surface hopping dynamics
