Marius WankoBCCMS

Excited State Photodynamics on Semiempirical and QM/MM Grounds

Marius Wanko

Bremen Center for Computational Material Science

Electron Dynamics in Complex Systems

Marius WankoBCCMS

Scientific Interest

• Focus: large chromophores embedded in a complex environment

• Optical properties (control by environment)

• Excited state dynamics FluorescenceFluorescenceproperties of GFP

Ultrafast/efficientPhotoisomerization of Retinal

Marius WankoBCCMS

Marius WankoBCCMS

Understanding the Photochemistry

1. Explore ground/excited states potential energy surface (PES)

2. Excited state molecular dynamics simulation• Sampling the configuration space, dynamic effects• State coupling/transitions in nonadiabatic regions• Decay to electronic ground state, product formation

Marius WankoBCCMS

Methodological Requirements

• ONIOM or QM/MM interface (point charges)

• Efficient QM method for ground/excited state energy (MD!)

• Analytic gradient, nonadiabatic-coupling vector,time propagation of density matrix ρij(t)

• Scheme for nonadiabatic dynamics

Challenges:• Correct description of entire relevant PES (!)• Correct response to environment (mechanical, electrostatic)

Marius WankoBCCMS

Surface Hopping Approach

• Semi-classical ansatz (classical propagation of nuclei)

• Nonadiabatic electronic wave function from numerical integrationof time-dependent Schrödinger/Kohn-Sham equation:

∑=ΨΨ∂−=Ψi

iit tcttitH );()(),;(),;(),;( elelel RrRrRrRr φh

( ) )(),(nintegratio num. ttciEcc ijij

ijijiji ρδ ⎯⎯⎯⎯ →⎯⋅−=∑ dR&h&

nonadiabaticcoupling vector

jiij φφ RRd ∇=)(

Marius WankoBCCMS

Surface Hopping Approach

Forces on nuclei

Ehrenfest: Surface hopping:),;(

el

tRR HRr

FΨ

∇=);(

el)(

RrF

iHRR

i

φ∇=

“Fewest switch” criterion [Tully1990]: hop only when occ. |ρii| changes!

• Stochastic representation of nuclear wave packet (non-coherent mixed state)

• Electronic coherence maintained

Marius WankoBCCMS

TD-DFTB Surface Hopping

2001 TD-DFTB nonadiabatic (Ehrenfest) MD + linear response2004 Analytical excited-state derivatives, density matrix

based on [Furche2002] (Z-vector equation):

0,,"'

xc3

xc"',

,, ≈=≈≈ ∑∑

σσσσσσ

α

αα

α

βα

βαβ

α

δρδρδρδγ EgqMqKqqK klij

Tklijklij

Sklij

2004 Davidson algorithmformal: O(N6) -> O(N4) scalingDFTB: N3.3 scaling

TDDFT TD-DFTB

System State Basis GS ES+∇ GS ES+∇

T-stilbene 1Bu SVP 411 1114 0.22 0.22

Hexahelicene 3A SV(P) 2050 2594 0.82 1.21

DMABN 1A1 TZVPP 4386 9056 0.23 0.32

Retinal 1A SVP 2148 7521 0.93 1.92

2006 NAC vectors, surface hopping

Marius WankoBCCMS

TD-DFTB Surface Hopping

• PES: good agreement with TDDFT methods (valence π-π* excitations)

• Convergence problems near CI -> unrestricted linear response

• NAC vector: 2-center approximation of Chernyak-Mukamelagreement with CASSCF

Marius WankoBCCMS

Failure of TDDFT

• Qualitative errors in PES from TDDFT(B) – static correlation (multi-configurational ground state)– open shell ground state in nonadiabatic region (CI)– charge-transfer (failure of local X-kernel in extended π-systems)– wrong CI topology

Correct PES and response to environment requires:– explicit treatment of exchange and static correlation– balanced incorporation of dynamic/static correlation

correct TDDFT(N-2)

Marius WankoBCCMS

Semiempirical MRCI

• Idea of MRCI/MRMP2:I. Small selection of configurations “reference” to catch essential

electronic structure of relevant states (static correlation)II. Include dynamic correlation (Coulomb hole) via CISD (MP2)

• Semiempirical: (II.) included in Fock matrix by definition!

• Implicit correlation + orthogonalization reduces 2-electron integrals (2-center: short-range damped , 3- and 4-center: negligible)

=> CI matrix much more sparse

Marius WankoBCCMS

Semiempirical MRCI

• Quick convergence wrt. active-orbital window and reference!

=> correlate only π-system, use small reference

• Same effect in Grimme’s DFT/MRCI (empirical scaling of 2-e integrals to avoid double counting of dynamic correlation)

3# reference configurations

Marius WankoBCCMS

OM2/MRCI

• Conventional MNDOs (AM1, PM3) fitted to ground state/equilibrium propertiespoor single-particle spectrum (too small HOMO-LUMO gap)

• OM1/OM2: new function for resonance integral consistent with shape of 2-center 1-e integrals in Löwdin-orthogonalized basis λHµν

=> improved MRCI excitation energies

Molecule State AM1 PM3 OM2 Exp.

HBDI (anion) 1A’ 2.08 2.08 2.56 2.59

Octatetraene 1Ag1Bu

3.304.43

2.984.30

4.374.87

3.974.41

OT-retinal (gas phase) La 1.84 1.79 2.33 2.03

Naphthalene 1B2u1B3u

4.042.28

3.822.44

4.873.96

4.664.13

Azulene 1B21A11B2

1.232.672.96

1.052.412.61

1.843.383.97

1.783.524.19

Marius WankoBCCMS

OM2/MRCI: Application to Spectral Tuning

• OM2/MRCI reproduces exp./ab initio for various kinds of spectral shifts:– Polarization (counter ion, entire protein environment)– Dihedral distortion of conjugated chain and ß-ionone ring– Bond-length alternation– bR vs. sRII, various mutants

Opsin shift bR–sRIIExp.: +0.32 eVOM2/MRCI: +0.31 eV

Shift Rho–BathoExp.: -0.21 eVOM2/MRCI: -0.19 eV

• Quantitative accuracy much better than CASSCF (or CIS)• Succeeds in cases where TDDFT fails (S1 gradient, charge transfer)

3 to go…

Marius WankoBCCMS

OM2/MRCI Surface Hopping

• Nonadiabatic time propagation• Tully’s minimal switching• Semi-analytical OM2 gradient• Scaling O(N4)• QM/MM-ready (included in CHARMM)• CI optimizer, state following, MO mapping

• Bug in gradient => only preliminary examples here

2 to go…

Marius WankoBCCMS

OM2/MRCI Surface Hopping

S0 population |ρ00|

EnsembleS1 occupation

1 to go…

Marius WankoBCCMS

Collaborations, Acknowledgement

• Michael Hoffmann (BCCMS Bremen)• Marcus Elstner, Jan Frähmcke, Prasad Fatak (TU Braunschweig)• Dávid Heringer (Budapest), Paul Strodel (Cambridge)

• Tom Keal, Axel Koslowski, Walter Thiel (MPI Mülheim)• Frank Neese (Universität Bonn)• Emad Tajkhorshid (Urbana-Champaign)• Peter König, Qiang Cui (Madison)

€: DFG Forschergruppe “Molecular mechanisms of Retinal protein action”