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Ultrafast processes in moleculesUltrafast processes in molecules
Mario [email protected]
XI – Surface hopping with correlated single reference methods
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Energy
Time
SR
ia SR
ja SR
SR
i ja a SR
i ja a SR
MR
MR
Dynamic Electron Correlation
Static EC
If DEC is needed everywhere and SEC only at the intersection, why choosing CASSCF (only SEC) for
dynamics?
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We need good SR method for Surface Hopping
SR can describe the relaxation through the manifold of excited states until:• The minimum of the first excited state is found (Kasha’s rule)• The crossing to the ground state is found
We can get information on:• Lifetimes• Reaction pathways distributions
We can’t get information on:• Photoisomerization quantum yield• Ground-state vibrational relaxation
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• Curchod, Rothlisberger, Tavernelli, ChemPhysChem 14, 1314 (2013)
TDDFT has been the SR choice for a number of groups:
• Bonačić-Koutecký, Mitrić• Tavernelli• Tapavicza, Furche• Ourselves
And about Coupled-Cluster?
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1 1 2 2 n n
TCC e HF
T t t t
• Sneskov and Christiansen, WIREs 2, 566 (2011)
Coupled-Cluster
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1 1 2 2 n n
TCC e HF
T t t t
i it
amplitudes
Excitation operator ith-order
• Sneskov and Christiansen, WIREs 2, 566 (2011)
Coupled-Cluster
ij
ab
2abij
ij
ab
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1 1 2 2 n n
TCC e HF
T t t t
• Sneskov and Christiansen, WIREs 2, 566 (2011)
• Choose a truncation level• Insert this Ansatz into TDSE• Get a set of nonlinear equations for the amplitudes and ground-state energy
i
T Ti
T TCC
e t e He HF
E HF e He HF
Coupled-Cluster
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1 1 2 2 n n
TCC e HF
T t t t
• Sneskov and Christiansen, WIREs 2, 566 (2011)
CCS < CCSD < CCSDT < ···
Truncation produces a well defined hierarchy of methods:
CC3, CCSD(T)
Coupled-Cluster
With some exotic flavors
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i
T Tie t e He HF
For excited states, the important quantity is the CC Jacobian matrix A:
i
i i
i
eA
t
Response Theory shows that:• Excited-state energies W are eigenvalues of A • Contribution R from each determinant is the eigenvector
The problem is to solveAR R
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i i
i i i i
i i
e eA A
t t
Jacobian is not a symmetric matrix!
1
AR R
LA L
LR
=====================================
Left and right eigenvalues converged to the same Results within 0.42E-07 a.u.
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CCS < CCSD < CC3 < CCSD(T) < CCSDT < ···
CC is time-consuming…
CC2CIS(D∞)
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• Schreiber, Silva-Junior, Sauer, Thiel, J Chem Phys 128, 134110 (2008)
CC is very accurate
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CC doesn’t work well for degenerated excited states
Non-symmetric Jacobian → eigenvalues (energies) may be imaginary
CCS < CIS(D∞ ) < CC2 < CCSD < CC3 < CCSD(T) < CCSDT < ···
CIS DA
†1ADC 2 CIS D CIS D
2
A A A
Build a symmetric Jacobian:
• Hättig, Köhn, J Chem Phys 117, 6939 (2002)
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CCS < CIS(D∞ ) < CC2 < CCSD < CC3 < CCSD(T) < CCSDT < ···
ADC(2)
ADC(2): Algebraic Diagrammatic Construction scheme up to second order
ADC(2) is not strictly a CC method
It is a excited-state propagator for ground-state MP2 (Schirmer1982)
These excited states are equivalent to the symmetrized CIS(D∞)
Trofimov, Krivdina, Weller, Schirmer, Chem Phys 329, 1 (2006)
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For surface hopping, we need nonadiabatic couplings Fkl
kl k l
F
R
Analytical nonadiabatic coupling vectors are available only for EOM-CC (CFour)
• Tajti and Szalay, J. Chem. Phys. 131, 124104 (2009)
*2
2max 0, Rel k k l kl
l
tP c c
c
F v
They are used to compute transition probabilities
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• Hammes-Schiffer and Tully, J Chem Phys 101, 4657 (1994)
In the 1990s, Hammes-Schiffer and Tully showed that
12 2
2kl k l k lt t t t t t t tt
F v
Overlap of electronic wavefunction in different time steps
*2
2max 0, Rel k k l kl
l
tP c c
c
F v
But we follow another way
For the probabilities, we don’t need F, we need F.v
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Couplings based on such overlaps have been used by several authors with:• MRCI• CASSCF• TDDFT• TDDFTB
• Barbatti, WIREs1, 620 (2011)
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Newton-XCIOVERLAP
modulus
MO coefficientsAO integrals for “Double-Molecule”CI coefficients
k lt t t
• Plasser, Granucci, Pittner, Barbatti, Persico, Lischka, J Chem Phys 137, 22A514 (2012)
The couplings can be used not only to get Fkl.vBut also to solve Surface Hopping equations with Local Diabatization method
12 2
2kl k l k lt t t t t t t tt
F v
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We use the eingenvectors R and L to build CIS wavefunctions
1
1
1
1
1
1
R kk
L kk
R
L
L Rk lk lt t t t t t
12 2
2kl k l k lt t t t t t t tt
F v
*2
2max 0, Rel k k l kl
l
tP c c
c
F v
ki ak ai
a i k
A aiE
For TDDFT (Casida 1996):
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Test case: adenine
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9H-adenine in the gas phase
Adenine is great for benchmarking:
a) Dynamics is available at:• MRCIS• OM2/MRCI• FOMO-CI/AM1• TDDFT (several functionals)• TDDFTB
b) Different dynamics methods have been used:• Surface Hopping• Ehrenfest Dynamics• Quantum wavepacket
c) Gas-phase transient spectra available for several pump wavelengths
• Barbatti, Lan, Crespo-Otero, Szymczak, Lischka, Thiel, J Chem Phys 137, 22A503 (2012)
N
N N
NH
NH2
2
6
9 S1 min
C2 CI C6 CI
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RI-ADC(2)/aug-cc-pVDZ
RI-CC2/aug-cc-pVDZ Expt.
E (eV)
f E (eV) f E (eV) f
n-p* 5.00 0.026 5.08 0.021p-p* 5.06 0.169 5.13 -0.034
5.16 ± 0.07 a 0.24 b
p-p* 5.07 0.101 5.16 0.300p-3s 5.39 0.010 5.39 0.013
Vertical excitation
a Barbatti and Ullrich, PCCP 13, 15492 (2011)b Clark, Peschel, Tinoco, J Phys Chem 69, 3615 (1965)
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CC2 ADC(2) Expt.Fluorescence (eV) 3.10 2.96 3.86, 3.00 a
Band origin (eV) 4.42 4.29 4.40-4.47 b
Fluorescence in water (f = 2.6×10-4)Band origin: vapor
S1 minimum
N
N N
NH
NH2
2
6
9 S1 min
C2 CI C6 CI
a Daniels and Hauswirth, Science 171, 675 (1971)b Nir, Plutzer, Kleinermanns, de Vries, Eur Phys J D 20, 317 (2002)
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ADC(2) CC2 Expta
Band Maximum 4.87 4.95 4.92FWHM 0.62 0.57 0.55smax (Å2.molecule-1) 0.50 0.53 0.46
4.0 4.5 5.00.0
0.2
0.4
0.6
CC2
Abs
orpt
ion
cros
s se
ctio
n (Å
2 .mol
ecul
e-1)
Expt.ADC(2)
400 350 300 250
Wavelength (nm)
4.0 4.5 5.00.0
0.2
0.4
0.6
M
Energy (eV)
ADC(2) (V) L
400 350 300 250
a Clark, Peschel, Tinoco, J Phys Chem 69, 3615 (1965)
Absorption
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Initial conditions were sampled in two spectral domains corresponding to different pump excitations
Different proportions of excitations into S1, S2 and S3 contribute to each domain
4.0 4.5 5.00.0
0.2
0.4
0.6
CC2
Abs
orpt
ion
cros
s se
ctio
n (Å
2 .mol
ecul
e-1)
Expt.ADC(2)
400 350 300 250
Wavelength (nm)
4.0 4.5 5.00.0
0.2
0.4
0.6
M
Energy (eV)
ADC(2) (V) L
400 350 300 250
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Simulations setup:
• 50 Trajectories in each domain (L and M)• Fewest Switches with decoherence correction• 0.5 fs time step for classical equations• 0.025 fs for quantum equations• Max 1000 fs or until E1-E0 < 0.1 eV• RI-CC2 and RI-ADC(2)• aug-cc-pVDZ• 3 excited states• Newton-X / Turbomole
1000 ps ADC(2) trajectory takes 24 days in 4 cores Xenon 2.7 GHz
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CC2 trajectories died in less than 100 fs!
Non-symmetrical Jacobian is the problem
ADC(2) trajectories are perfectly stable
They ran until one of the termination criteria was satisfied
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0.2
0.3
0.4
0.5
0.0 0.2 0.4 0.6 0.8 1.0
0.04
0.06
0.08
0.10
0.0 0.2 0.4 0.6 0.8 1.0
D2
diagn
ostic
for S
1
Low Medium
D1
diagn
ostic
for S
0
Time (ps) Time (ps)
SR Ground state
MR Ground state
Single excitation
Double excitation
• Nielsen and Janssen, Chem Phys Lett 310, 568 (1999)
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Domain SH/ADC(2) Expt.a
L 0.50 ± 0.12 0.62 ± 0.02M 0.57 ± 0.12 0.68 ± 0.02
a Evans and Ullrich, J Phys Chem A, 114, 11225 (2010)
Deactivation to Ground State within 1 ps
0.9
(1 ) 0.57(1 0.57)1.645 0.12
50M
p pZ
N
Margin of error for 90% confidence interval
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Domain SH/ADC(2) Expt.a
L 0.50 ± 0.12 0.62 ± 0.02M 0.57 ± 0.12 0.68 ± 0.02
a Evans and Ullrich, J Phys Chem A, 114, 11225 (2010)
Deactivation to Ground State within 1 ps
0.9
(1 ) 0.57(1 0.57)1.645 0.12
50M
p pZ
N
Margin of error for 90% confidence interval
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Domain SH/ADC(2) Expt.a
L 0.50 ± 0.12 0.62 ± 0.02M 0.57 ± 0.12 0.68 ± 0.02
a Evans and Ullrich, J Phys Chem A, 114, 11225 (2010)
Deactivation to Ground State within 1 ps
• There is no statistical distinction between L and M• SH/ADC(2) is slightly underestimating the deactivation level Problems with ADC(2) surfaces? Expt. includes 9H and 7H tautomers Expt. includes ionization info (Barbatti and Ullrich 2011)
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N
N N
NH
NH2
2
6
9 S1 min
C2 CI C6 CI
Trajectories cluster around C2 deformation of the pyrimidine ring
C2
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Domain C2 C6 H-elimin.L 0.52 0.36 0.12M 0.54 0.43 0.03
Participation of each reaction path in the internal conversion
• C2 is the dominant one• C6 is also important• H elimination plays a minor role
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0 2 4 6 83.0
3.5
4.0
4.5
5.0
Ene
rgy
(eV
)
Mass-weighted distance (Å.amu1/2)
S1 min
C2 CIC6 CI
There are kinetic reasons favoring C2There are thermodynamic reasons favoring C2
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Exper
imen
tal
ADC(2)
MRCIS
OM2/MRCI
TD-P
BE
TD-B
97XD
TD-B
3LYP
TD-P
BE0
TD-C
AM-B
3LYP
TD-B
HLYP
TD-M
06-H
F
0
20
40
60
80
S0 p
opul
atio
n at
1 p
s (%
)
C2 puckering C6 puckering H elimination Experimental682
5712
858
598
58128
2015
0
20212516
1016
• Only ADC(2) and OM2/MRCI predict right IC• MRCIS overshoots IC• TDDFT underestimates IC
• OM2/MRCI underestimates C2
• ADC(2) is the best result so far
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To conclude:
• SR methods can be very useful for some (not all!) problems in nonadiabatic dynamics• Non-symmetric Jacobians in CC methods are a major problem• ADC(2) showed good potential (accurate and stable)• Lack of hopping to ground state is the main problem