Ultrafast processes in molecules

Ultrafast processes in molecules

Mario [email protected]

XI – Surface hopping with correlated single reference methods

2

Energy

Time

SR

ia SR

ja SR

SR

i ja a SR

i ja a SR

MR

MR

Dynamic Electron CorrelationStatic EC

If DEC is needed everywhere and SEC only at the intersection, why choosing CASSCF (only SEC) for

dynamics?

3

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

4• 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?

5

1 1 2 2 n n

TCC e HF

T t t t

• Sneskov and Christiansen, WIREs 2, 566 (2011)

Coupled-Cluster

6

1 1 2 2 n n

TCC e HF

T t t t

i it

amplitudes

Excitation operator ith-order


Coupled-Cluster

ijab

2abij

ijab

7

1 1 2 2 n n

TCC e HF

T t t t


• 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

8

1 1 2 2 n n

TCC e HF

T t t t


CCS < CCSD < CCSDT < ···Truncation produces a well defined hierarchy of methods:

CC3, CCSD(T)

Coupled-Cluster

With some exotic flavors

9

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 solveWAR R

10

i i

i i i i

i i

e eA A

t t

Jacobian is not a symmetric matrix!

1

W W

AR RLA LLR

=====================================

Left and right eigenvalues converged to the same Results within 0.42E-07 a.u.

11

CCS < CCSD < CC3 < CCSD(T) < CCSDT < ···CC is time-consuming…

CC2CIS(D∞)

12• Schreiber, Silva-Junior, Sauer, Thiel, J Chem Phys 128, 134110 (2008)

CC is very accurate

13

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 D2

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)

15

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 kll

tP c cc

F v

They are used to compute transition probabilities

16• Hammes-Schiffer and Tully, J Chem Phys 101, 4657 (1994)

In the 1990s, Hammes-Schiffer and Tully showed that

1 2 22kl k l k lt t t t t t t tt

F v

Overlap of electronic wavefunction in different time steps

*2


tP c cc

F v

But we follow another way

For the probabilities, we don’t need F, we need F.v

17

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


F v

19

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


F v

*2


tP c cc

F v

ki ak ai

a i k

A aiE

For TDDFT (Casida 1996):

20

Test case: adenine

21

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

22

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 bp-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)

23

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

Abso

rptio

n cr

oss s

ectio

n (Å

2 .mole

cule-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

Abso

rptio

n cr

oss s

ectio

n (Å

2 .mole

cule-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 diag

nostic for S

1

Low Medium

D1 diag

nostic 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.aL 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

50Mp pZN

Margin of error for 90% confidence interval

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0.9(1 ) 0.57(1 0.57)1.645 0.12

50Mp pZN

Margin of error for 90% confidence interval

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• 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)

32

N

N N

NH

NH2

2

6

9 S1 min

C2 CI C6 CI

Trajectories cluster around C2 deformation of the pyrimidine ringC2

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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

5.0En

ergy

(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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Experim

ental

ADC(2)MRCIS

OM2/MRCI

TD-PBE

TD-B

97XD

TD-B3LY

P

TD-PBE0

TD-CAM-B3LY

P

TD-BHLY

P

TD-M06-

HF0

20

40

60

80

S 0 p

opula

tion

at 1

ps (

%)

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

Ultrafast processes in molecules

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