Reminder from last lecture: potential energy surfaces

Transcription

1 Outline 1 Why Quantum Dynamics? A TDDFT approach Ivano Tavernelli IBM -Research Zurich Summer School of the Max-Planck-EPFL Center for Molecular Nanoscience & Technology 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory 1 Why Quantum Dynamics? Reminder from last lecture: potential energy surfaces 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei...

2 Reminder from last lecture: potential energy surfaces Why Quantum dynamics? Why Quantum Dynamics? GS adiabatic dynamics (BO vs. CP) Energy BO M I R I (t) = r min E KS ({ i [ ]}) CP µ i i (t)i = h i EKS ({ i (r)}) + h i {constr.} M I R I (t) = re KS ({ i (t)}) ES nonadiabatic quantum dynamics Wavepacket dynamics (MCTDH) Energy Trajectory-based approaches - Tully s trajectory surface hopping (TSH) - Bohmian dynamics (quantum hydrodyn.) - Semiclassical (WKB, DR) - Path integrals (Pechukas) - Mean-field solution () We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei... Density matrix, Liouvillian approaches,... Why Quantum Dynamics? Why Quantum Dynamics? Why Quantum dynamics? Why Quantum dynamics? GS adiabatic dynamics GS adiabatic dynamics Energy First principles Heaven Ab initio MD with WF methods Ab initio MD with DFT & TDDFT [CP] classical MD Coarse-grained MD... No principles World Energy First principles Heaven Ab initio MD with WF methods Ab initio MD with DFT & TDDFT [CP] classical MD Coarse-grained MD... No principles World ES nonadiabatic quantum dynamics ES nonadiabatic quantum dynamics Energy First principles Heaven Ab initio MD with WF methods Ab initio MD with DFT & TDDFT [CP] # Models #? No principles World Energy (-) We cannot get read of electrons (-) Nuclei keep some QM flavor (-) Accuracy is an issue (-) Size can be large (di use excitons) (+) Time scales are usually short (< ps)

3 1 Why Quantum Dynamics? 2 Nonadiabatic Bohmian dynamics Nonadiabatic e ects requires quantum nuclear dynamics The nuclear dynamics cannot be described by a single classical trajectory (like in the ground state -adiabatically separated- case) 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory Multi-trajectories vs. wavepacket Wavepacket-dynamics vs. trajectory-based QD Trajectory-based dyn. Wavepacket dyn. Trajectory-based dynamics Wavepacket-based dynamics dynamics in phase space d=3n-6 (unbiased) reaction coordinates d=1-10 (knowledge based) on-the-fly dynamics pre-computed energy surfaces on a grid good scaling (linear in d) scaling catastrophe (M d ) coupling to real environment coupling to harmonic bath slow convergence good convergence approximate exact in low dimensions Dynamics in 3N dimensions Approximated quantum dynamics (trajectory ensemble) Dynamics in 8dimensions Exact quantum dynamics (nuclear wavepakets)

4 Wavepacket dynamics with MCTDH Wavepacket dynamics with MCTDH The multiconfiguration time-dependent Hartree, MCTDH, approach. 1) collecting variables: Precomputed TDDFT surfaces along selected modes (8). Potential energy surfaces computed along 2 vibrational modes (Spin Orbit Coupling (SOC) described by the Breit-Pauli operator.) (R 1, R 2,...,R 3N )! (Q 1,...,Q M ) with M << 3N, and Q 1 (R 1, R 2,...,R 3N )... Q M (R 1, R 2,...,R 3N ) 2) Ansatz: linear combination of Hartree products n 1 n f (Q 1, Q 2,...,Q M )= A j1...j f (t) j 1 =1 j f =1 3) EOF from Dirac-Frenkel variational principle: h Ĥ i =0. fy k=1 (k) j k (Q k, t) state sym. ev f S1 B S2 A S3 B S4 B state sym. ev f T1 A T2 A T3 B T4 B mode cm 1 sym. type B3 Rocking B3 Jahn Teller B3 As. Stretching B3 Rocking B3 Rocking B3 Rocking A Twisting SOC at equilibrium: S 1 /T 1 =65cm 1 ; S 2 /T 1 =35cm 1 SOC at distorted geometry: S 1 /T 1 =30cm 1 ; S 2 /T 1 =30cm 1 Wavepacket-dynamics of Cu-phenanthroline Wavepacket-dynamics of Cu-phenanthroline 0fs 20 fs

5 Wavepacket-dynamics of Cu-phenanthroline Wavepacket-dynamics of Cu-phenanthroline 100 fs 500 fs Trajectory-based quantum dynamics Trajectory-based quantum dynamics: an example Ultrafast tautomerization of 4-hydroxyquinoline (NH 3 ) 3 Hydrogen or proton transfer? CIS/CASSCF for the in-plane geometry! / crossing leads to a hydrogen atom transfer. S. Leutwyler et al., Science, 302, 1736(2003)

6 Trajectory-based quantum dynamics: an example Trajectory-based quantum dynamics: an example Ultrafast tautomerization of 4-hydroxyquinoline (NH 3 ) 3 Hydrogen or proton transfer? Ultrafast tautomerization of 4-hydroxyquinoline (NH 3 ) 3 Hydrogen or proton transfer? What about TDDFT combined with nonadiabatic dynamics (1)? What about TDDFT combined with nonadiabatic dynamics (2)? Trajectory-based quantum dynamics: an example Trajectory-based quantum dynamics: an example Ultrafast tautomerization of 4-hydroxyquinoline (NH 3 ) 3 Hydrogen or proton transfer? With TDDFT, we observe: Ultrafast tautomerization of 4-hydroxyquinoline (NH 3 ) 3 Hydrogen or proton transfer? Similar observations with CASPT2 calculations: Symmetry breaking. No crossing with the state. Proton transfer instead of a hydrogen transfer. Guglielmi et al., PCCP,11,4549(2009). Forcing in-plane symmetry: hydrogen transfer. Unconstrained geometry optimization leads to a proton transfer! Fernandez-Ramos et al., JPCA,111,5907(2007).

7 Summary: Why trajectory-based approaches? W1 In conventional nuclear wavepacket propagation potential energy surfaces are needed. W2 Di culty to obtain and fit potential energy surfaces for large molecules. W3 Nuclear wavepacket dynamics is very expensive for large systems (6 degrees of freedom, 30 for MCTDH). Bad scaling. T1 Trajectory based approaches can be run on-the-fly (no need to parametrize potential energy surfaces). T2 Can handle large molecules in the full (unconstraint) configuration space. T3 They o er a good compromise between accuracy and computational e ort. 1 Why Quantum Dynamics? 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory Starting point The starting point is the molecular time-dependent Schrödinger equation: Tarjectory-based quantum and mixed QM-CL solutions We can derive the following trajectory-based solutions: (r, R, t) =Ĥ (r, R, t) Nonadiabatic dynamics I. Tavernelli et al., Mol. Phys., 103, (2005). where Ĥ is the molecular time-independent Hamiltonian and (r, R, t) thetotalwavefunction (nuclear + electronic) of our system. Adiabatic Born-Oppenheimer MD equations In mixed quantum-classical dynamics the nuclear dynamics is described by a swarm of classical trajectories (taking a partial limit ~! 0forthenuclearwf). In this lecture we will discuss two main approximate solutions based on the following Ansätze for the total wavefucntion (r, R, t) Born-! Huang 1 j j (r; R) j (R, t) Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT, U. Rothlisberger, PCCP, 13, (2011) Nonadiabatic (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88, (2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95, (2005) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98, (2007) (r, R, t) Ehrenfest! apple i (r, t) (R, t)exp ~ Z t t 0 E el (t 0 )dt 0 Time dependent potential energy surface approach based on the exact decomposition: (r, R, t) = (R, t) (r, t). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105, (2010)

8 - the nuclear equation (r, R, t) Ehrenfest! apple Z i t (r, t) (R, t)exp E el (t 0 )dt 0 ~ t 0 We start from the polar representation of the nuclear wavefunction Inserting this representation of the total wavefunction into the molecular td Schrödinger equation and multiplying from the left-hand side by (R, t) andintegratingoverr we (r, t) i~ = where ˆV (r, R) = P e 2 i<j r i r j In a similar way, multiplying by ~ 2 applez r 2 i (r, t) + dr (R, t) ˆV (r, R) (R, t) (r, t) 2m e i P e 2 Z,i R r i. (r, t) andintegratingoverr we (R, t) i~ = ~2 applez M 1 r 2 (R, t) + dr 2 (r, t)ĥ el (r, t) (R, t) Conservation of energy has also to be imposed through the condition that dhĥi/dt 0. Note that both the electronic and nuclear parts evolve according to an average potential generated by the other component (in square brakets). These average potentials are time-dependent and are responsible for the feedback interaction between the electronic and nuclear components. apple i (R, t) =A(R, t)exp S(R, t) ~ where the amplitude A(R, t) and the phase S(R, t)/~ are real functions. Inserting this representation for (R, t) andseparatingtherealandtheimaginarypartsonegets for the phase S in the classical limit ~! = 1 applez M 1 r S 2 dr 2 (r, t)ĥ el (r, R) (r, t) This has the form of the Hamilton-Jacobi (HJ) equation of classical mechanics, which establishes a relation between the partial di erential equation for S(R, t) inconfigurationspace and the trajectories of the corresponding (quantum) mechanical systems. - the nuclear equation - the nuclear = 1 applez M 1 r S 2 dr 2 (r, t)ĥ el (r, R) (r, t) The identification of S(R, t) withthe classical action,definesapoint-particledynamicswith Hamiltonian, H cl and momenta P = r R S(R). The solutions of this Hamiltonian system are curves (characteristics) in the(r, t)-space, which are extrema of the action S(R, t) forgiveninitialconditionsr(t 0 )andp(t 0 )=r R S(R) R(t0 ). Newton-like equation for the nuclear trajectories corresponding to the HJ equation dp dt = r applez dr (r, t)ĥ el (r, R) (r, t) Instead of solving the field equation for S(R, t), find the equation of motion for the corresponding trajectories (r; R, t) i~ = Ĥ el (r; R) (r; R, t) M I R I = r I hĥ el (r; R)i

9 - the nuclear equation - Example The identification of S(R, t) withthe classical action,definesapoint-particledynamicswith Hamiltonian, H cl and momenta P = r R S(R). k(r, t) = 1 2m e r 2 r k(r, t)+v e [, 0](r, t) k (r, t) M I R I = r I hĥ el (r; R)i The solutions of this Hamiltonian system are curves (characteristics) in the(r, t)-space, which are extrema of the action S(R, t) forgiveninitialconditionsr(t 0 )andp(t 0 )=r R S(R) R(t0 ). Newton-like equation for the nuclear trajectories corresponding to the HJ equation dp dt = r applez dr (r, t)ĥ el (r, R) (r, t) - Densityfunctionalization ( k :KSorbitals) k(r, t) = 1 2m e r 2 r k(r, t)+v e [, 0](r, t) k (r, t) M I R I = r I E[ (r, t)] and mixing of electronic states and mixing of electronic (r; R, t) i~ = Ĥ el (r; R) (r; R, t) M I R I = r I hĥ el (r; R)i i~ċ k (t) =c k (t)e el k i~ j c j (t)d kj Consider the following expansion of (r; R, t) inthestatic basis of electronic wavefucntions { k (r; R)} 1 (r; R, t) = c k (t) k (r; R) k=0 The time-dependency is now on the set of coe cients {c k (t)} ( c k (t) 2 is the population of state k). Inserting in the Ehrenfest s equations... where M I R I = r I 1 D kj = h j i = h Thus we incorporate directly nonadiabatic e ects. k=0 c k (t) 2 E j i = Ṙh k r j i = Ṙ d kj

10 : the mean-field potential Tarjectory-based quantum and mixed QM-CL solutions We can derive the following trajectory-based solutions: Nonadiabatic dynamics I. Tavernelli et al., Mol. Phys., 103, (2005). Adiabatic Born-Oppenheimer MD equations Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT, U. Rothlisberger, PCCP, 13, (2011) i~ċ k (t) =c k (t)e el k i~ j c j (t)d kj Nonadiabatic (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88, (2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95, (2005) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98, (2007) M I R I = r I 1 k=0 c k (t) 2 E el k Time dependent potential energy surface approach based on the exact decomposition: (r, R, t) = (R, t) (r, t). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105, (2010) Born-Oppenheimer approximation Born-Oppenheimer approximation (r, R, t) Born-! Huang 1 j j (r; R) j (R, t) (r, R, t) Born-! Huang 1 Electrons are static. Useyourfavoriteel.str.method. j j (r; R) j (R, t) In this equation, j (r; R) describes a complete basis of electronic states solution of the time-independent Schrödinger equation: Ĥ el (r; R) j (r; R) =E el,j (R) j (r; R) R is taken as a parameter. Eigenfunctions of Ĥ el (r; R) areconsideredtobeorthonormal,i.e.h j i i = ij. For the nuclei, insertthisansatz into the molecular time-dependent Schrödinger equation After left multiplication by used h j i i = ij ): Ĥ (r, R, t) (r, R, t) k (r; R) andintegrationoverr, weobtainthefollowingequation(we " # ~ 2 1 r 2 I + E el,k (R) k (R, t)+ D kj j (R, t) 2M I I k(r, t) j

11 Born-Oppenheimer approximation Born-Oppenheimer approximation " # ~ 2 r 2 I + E el,k (R) k (R, t)+ 2M I I j Equation for the nuclear wavepacket, (R, t), dynamics. E el,k (R) representsapotentialenergysurfaceforthenuclei. D kj j (R, t) k(r, t) Important additional term : D kj! NONADIABATIC COUPLING TERMS Z D kj = + I " k (r; R) I Z 1 M I # ~ 2 r 2 I 2M I j (r; R)dr k (r; R)[ i~r I ] j (r; R)dr [ i~r I ] Z D kj = + I " k (r; R) I Z 1 M I # ~ 2 r 2 I 2M I j (r; R)dr k (r; R)[ i~r I ] j (r; R)dr [ i~r I ] If we neglect all the D kj terms (diagonal and o -diagonal), we have the Born-Oppenheimer approximation. " # ~ 2 r 2 I + E el,k (R) k (R, t) 2M I I k(r, t) Mainly for ground state dynamics or for dynamics on states that do not couple with others. (Back to nonadiabatic dynamics later). Born-Oppenheimer approximation: the nuclear trajectories Born-Oppenheimer approximation: the nuclear trajectories " # ~ 2 r 2 I + E el,k (R) k (R, t) 2M I I k(r, t) Using a polar expansion for k (R, t), we may find a way to obtain classical equation of motions for the nuclei. apple i k (R, t) =A k (R, t)exp ~ S k(r, t). A k (R, t) representsanamplitudeands k (R, t)/~ aphase. Further: insert the polar representation into the equation above, do some algebra, and separate real and imaginary part, we obtain an interesting set of k = ~2 2 I = I M 1 r 2 I A k 1 I A k 2 I M 1 I r I S k 2 E k M 1 1 I r I A k r I S k M 1 I A k r 2 I 2 S k I Dependences of the functions S and A are omitted for clarity (k is a index for the electronic state; in principle there is only one state in the adiabatic case). We have now a time-dependent equation for both the amplitude and the phase. Since we are in the adiabatic case there is only one PES and the second equation becomes trivially a di usion continuity equation. The nuclear dynamics is derived from the real part k ). This equation has again the form of a classical Hamilton-Jacobi equation.

12 Born-Oppenheimer approximation: the nuclear trajectories Born-Oppenheimer approximation: the nuclear k = ~2 2 I = I M 1 r 2 I A k 1 I A k 2 I M 1 I r I S k 2 E k M 1 1 I r I A k r I S k M 1 I A k r 2 I 2 S k k = ~2 2 I M 1 r 2 I A k 1 I A k 2 The classical limit is obtained by taking 1 : ~! k = 1 2 I M 1 I r I S k 2 I M 1 I r I S k 2 These are the classical Hamilton-Jacobi equation and S is the classical action related to a particle. E k E k The momentum of a particle I is related to Z t Z t S(t) = L(t 0 )dt 0 = Ekin (t 0 ) E pot(t 0 ) dt 0 t 0 t 0 Instead of solving the field equation for S(R, t), find the equation of motion for the corresponding trajectories (characteristics). r I S = p I = v I M I 1 Caution! This classical limit is subject to controversy... Born-Oppenheimer approximation: the nuclear trajectories Mean-field vs. BO MD (adiabatic case) Therefore, taking the gradient, r k = 1 2 r J I M 1 I r I S k 2 + rj E k and rearranging this equation using r J S k /M J = v k J, we obtain the (familiar) Newton equation: M J d dt v k J = r JE (r; R, t) i~ = Ĥ el (r; R) (r; R, t) M I R I = r I hĥ el (r; R)i Explicit time dependence of the electronic wavefunction. In Summary: Adiabatic BO MD Ĥ el (r; R) M I RI = k (r; R) =E el k (R) k(r; R) r I E el k (R) = r I h k Ĥ el ki min k Born-Oppenheimer dynamics Ĥ el (r; R) M I RI = k (r; R) =E el k (R) k(r; R) r I E el k (R) = r I h k Ĥ el ki min k The electronic wavefunction are static (only implicit time-dependence).

13 Mean-field vs. BO MD (adiabatic case) Tarjectory-based quantum and mixed QM-CL solutions We can derive the following trajectory-based solutions: Method Born-Oppenheimer MD Ehrenfest MD adiabatic MD (one PES) nonadiabatic MD (mean-field) E cient propagation of the nuclei Get the real dynamics of the electrons Adiabatic nuclear propagation Propagation of nuclei & electrons t a.u. ( fs) t 0.01 a.u. (0.25 as) Simple algorithm Common propagation of the nuclei and the electrons implies more sophisticated algorithms Exact quantum dynamics? Can we derive exact quantum equations of motion for the nuclei? (without taking the classical limit ~! 0?) Nonadiabatic dynamics I. Tavernelli et al., Mol. Phys., 103, (2005). Adiabatic Born-Oppenheimer MD equations Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT*, U. Rothlisberger, PCCP, 13, (2011) (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88, (2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95, (2005) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98, (2007) Time dependent potential energy surface approach based on the exact decomposition: (r, R, t) = (R, t) A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105, (2010) (r, t). Nonadiabatic Bohmian dynamics Nonadiabatic Bohmian dynamics NABDY: exact trajectory-based nonadiabatic dynamics Using (r, R, t) = P 1 j j (r; R) j (R, t) j (R, t) =A j (R, t)exp i ~ Sj (R, t) in the exact time-dependent Schrödinger equation for the nuclear wavefucntion we get j (R, t) = 1 j (R, t) = r S j (R, t) 2 + Ej el (R) ~ 2 2M r 2 A j (R, t) A j (R, t) + ~ 2 D (R) A i (R, t) < he i i ~ 2 d (R) r A i (R, t) < he i i i 2M ji A j (R, t),i6=j M ji A j (R, t) + ~ d (R) A i (R, t) r,i6=j M ji A j (R, t) 1 r A j (R, t)r S j (R, t) M ~ + i 2M S i (R, t)= he i i 1 2M D (R)A ji i (R, t)= he i i,i6=j 1 d (R)A,i6=j M ji i (R, t)r S i (R, t)< he i i, A j (R, t)r 2 S j (R, t) ~ d (R)r M ji where both S j (R, t) anda j (R, t) arerealfieldsand = 1 ~ (Si (R, t) Sj (R, t)). A i (R, t)= he i i NABDY: exact trajectory-based nonadiabatic dynamics From the NABDY equations we can obtain a Newton-like equation of motion (using the HJ definition of the momenta r S j (R, t) =P j ) M d 2 R h (dt j ) 2 = r E j el (R)+Q j (R, t)+ i D ij (R, t) i where Q j (R, t) isthequantumpotentialresponsibleforallcoherence/decoherence intrasurface QM e ects, and D j (R, t) isthenonadiabatic potential responsible for the amlpitude transfer among the di erent PESs. For more informations see: B. Curchod, IT, U. Rothlisberger, PCCP, 13, (2011) NABDY limitations Many numerical challenges Instabilities induced by the quantum potential Compute derivatives in the 3N dimensional(r 3N )configurationspace

14 Nonadiabatic Bohmian dynamics Tarjectory-based quantum and mixed QM-CL solutions We can derive the following trajectory-based solutions: Nonadiabatic dynamics I. Tavernelli et al., Mol. Phys., 103, (2005). Adiabatic Born-Oppenheimer MD equations Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT, U. Rothlisberger, PCCP, 13, (2011) (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88, (2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95, (2005) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98, (2007) Time dependent potential energy surface approach based on the exact decomposition: (r, R, t) = (R, t) A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105, (2010) (r, t). TSH nonadiabatic MD There is no derivation of TSH dynamics. The fundamental hypothesis beyond TSH is that it is possible to design a dynamics that consists of: propagation of a quantum amplitude, Ck (t), associated to each PES, k 1 (r, R, t) = Ck (t) k(r; R) k (the label is to recall that we have a di erent contribution from each di erent trajectory). classical (adiabatic) timeevolutionofthenucleartrajectoriesonadiabaticstatessolution of the Schrödinger equation for the electronic sub-system. Each trajectory carries a phase, Ck (t), associated to each surface k. transitions (hops) of the trajectories between electronic states according to a stochastic algorithm, which depends on the nonadiabatic couplings and the amplitudes Ck (t). See also: J. Tully, Faraday discussion, 110, 407(1998). Tully s surface hopping - How does it work? Tully s surface hopping - How does it work? The main claim of TSH is that, the collection of a large enough set of independent trajectories gives an accurate representation of the nuclear wave packet CL k (R, t )= N k (R, dv, t ) 1 N tot dv k(r, t ) 2 Ck,R,t 2 Equation of motion for the amplitudes: Switching algorithm: i~ C k (t) = j C j (t)(h kj i~ṙ d kj ) Inserting (r, R, t) = 1 Ck (t) k(r; R) into the molecular time-dependent Schrödinger equation and after some rearrangement, we obtain: i~ C k (t) = j k C j (t)(h kj i~ṙ d kj ) with H kj = h k (r; R)i Ĥ el j (r; R)i. In the adiabatic representation, we have H kk = E el k and H kj =0. In the fewest switches algorithm, the transition probability from state j to state k in the time interval [t, t + dt] is given by: Z t+dt gjk (t, t + dt) 2 d =[C k t ( )C j H kj ( )] <[Ck ( )C C j where kj ( ) =Ṙ d kj ( ). A hop occurs between j and k if lapplek 1 g jl < < lapplek where is generated randomly in the interval [0, 1]. g jl, j ( ) ( )C j ( ) kj ( )]

15 Tully s surface hopping - Summary Tully s surface hopping Tully s surface hopping - Examples 1D systems i~ C k (t) = j C j (t)(h kj i~ṙ d kj ) M I RI = gjl < < lapplek 1 lapplek r I E el k (R) g jl, Some warnings: 1 Evolution of classical trajectories (no QM e ects such as tunneling are possible). 2 Rescaling of the nuclei velocities after a surface hop (to ensure energy conservation) is still amatterofdebate. 3 Depending on the system studied, many trajectories could be needed to obtain a complete statistical description of the non-radiative channels. For more details (and warnings) about Tully s surface hopping, see G. Granucci and M. Persico, JChemPhys126,134114(2007). J.C. Tully, J. Chem. Phys. (1990), 93, 1061 Trajectory-based Very good nonadiabatic agreement molecular withdynamics exact nuclear wavepacket propagation. Tully s surface hopping - Examples 1D systems Tully s surface hopping - Examples 1D systems J.C. Tully, J. Chem. Phys. (1990), 93, 1061 J.C. Tully, J. Chem. Phys. (1990), 93, 1061 On the right: population of the upper state (k=mom) exact TSH Landau-Zener Trajectory-based Very good nonadiabatic agreement molecular withdynamics exact nuclear wavepacket propagation. Trajectory-based Very good nonadiabatic agreement molecular withdynamics exact nuclear wavepacket propagation.

16 Tully s surface hopping - Examples 1D systems Comparison with wavepacket dynamics Butatriene molecule: dynamics of the radical cation in the first excited state. J.C. Tully, J. Chem. Phys. (1990), 93, 1061 On the right: population of the upper state exact TSH JPCA,107,621 (2003) Trajectory-based Very good nonadiabatic agreement molecular withdynamics exact nuclear wavepacket propagation. Comparison with wavepacket dynamics Comparison with wavepacket dynamics Butatriene molecule: dynamics of the radical cation in the first excited state. Nuclear wavepacket dynamics on fitted potential energy surfaces (using MCTDH with 5 modes). Reappearing of the wavepacket in S 1 after 40fs. JPCA,107,621 (2003) JPCA,107,621 (2003) CASSCF PESs for the radical cation (Q 14 :symmetricstretch, : torsionalangle).

17 Comparison with wavepacket dynamics Comparison with wavepacket dynamics On-the-fly dynamics with 80 trajectories (crosses). On-the-fly dynamics with 80 trajectories. Trajectories are not coming back close to the conical intersection. Trajectories are not coming back close to the conical intersection. What is the reason for this discrepancy? The independent trajectory approximation?, i.e. the fact that trajectories are not correlated? (Or it has to do with di erences in the PESs?) What is the reason for this discrepancy? The independent trajectory approximation?, i.e. the fact that trajectories are not correlated? (Or it has to do with di erences in the PESs?) JPCA,107,621 (2003) JPCA,107,621 (2003) TDDFT-based trajectory surface hopping 1 Why Quantum Dynamics? 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory TDDFT-based trajectory surface hopping Tully s surface hopping - On-the-fly dynamics Tully s surface hopping i~ C k (t) = j M I RI = lapplek 1 g jl C j (t)(h kj i~ṙ d kj ) r I E el k (R) < < lapplek g jl, What about the electronic structure method for on-the-fly dynamics? We need: Potential energy surfaces! MR-CISD, LR-TDDFT, semiempirical,... Forces on the nuclei! MR-CISD, LR-TDDFT, semiempirical methods,.... Nonadiabatic coupling terms! MR-CISD, LR-TDDFT (?), semiempirical methods,....

18 TDDFT-based trajectory surface hopping Tully s surface hopping - On-the-fly dynamics TDDFT-based trajectory surface hopping Tully s surface hopping - On-the-fly dynamics Tully s surface hopping Tully s surface hopping i~ C k (t) = j C j (t)(h kj i~ṙ d kj ) i~ C k (t) = j C j (t)(h kj i~ṙ d kj ) M I RI = r I E el k (R) M I RI = r I E el k (R) lapplek 1 g jl < < lapplek g jl, lapplek 1 g jl < < lapplek g jl, What about the electronic structure method for on-the-fly dynamics? We need: Potential energy surfaces! MR-CISD, LR-TDDFT, semiempirical,... Forces on the nuclei! MR-CISD, LR-TDDFT, semiempirical methods,.... Nonadiabatic coupling terms! MR-CISD, LR-TDDFT (?), semiempirical methods,.... What about the electronic structure method for on-the-fly dynamics? We need: Potential energy surfaces! MR-CISD, LR-TDDFT, semiempirical,... Forces on the nuclei! MR-CISD, LR-TDDFT, semiempirical methods,.... Nonadiabatic coupling terms! MR-CISD, LR-TDDFT (?), semiempirical methods,.... TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT TDDFT-TSH: Applications Nonadiabatic couplings with LR-TDDFT? Nonadiabatic coupling vectors are defined in terms of electronic wavefunctions: d kj = h k (R) r R j (R)i = h k(r) r R Ĥ el j (R)i E j (R) E k (R) The main challenge is to compute all these quantities as a functional of the ground state electronic density (or equivalently, of the occupied Kohn-Sham orbitals). d kj! d kj [ ] Di erent approaches for the calculation of d 0j [ ] are available 2. Here we will use the method based on the auxiliary many-electron wavefunctions. 2 V. Chernyak and S. Mukamel, J. Chem. Phys. 112, 3572 (2000); R. Baer, Chem. Phys. Lett. 364, 75 (2002); E. Tapavicza, I. Tavernelli, and U. Rothlisberger, Phys. Rev. Lett. 98, (2007); C. P. Hu, H. Hirai, and O. Sugino, J. Chem. Phys. 127, (2007). 1 Why Quantum Dynamics? 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 5 Local control (LC) theory

19 TDDFT-TSH: Applications Protonated formaldimine The protonated formaldimine is a model compound for the study of isomerization in rhodopsin chromophore retinal. In addition to the ground state (GS), two excited electronic states are of interest: 1 S 1 :! (low oscillator strength) 2 S 2 :! (high oscillator strength) TDDFT-TSH: Applications Protonated formaldimine Computational details Isolated system LR-TDDFT/PBE/TDA SH-AIMD 50 trajectories (NVT) each of 100 fs. PRL, 98, (2007); THEOCHEM, 914, 22 (2009). TDDFT-TSH: Applications Protonated formaldimine Protonated formaldimine as a model compound for the study of the isomerization of retinal. TDDFT-TSH: Applications Protonated formaldimine Typical trajectory Photo-excitation promotes the system mainly into S 2. Relaxation involves at least 3 states: S 0 (GS), S 1 and S 2. [E. Tapavicza, I. T., U. Rothlisberger, PRL, 98, (2007); THEOCHEM, 914, 22 (2009)] PRL, 98, (2007); THEOCHEM, 914, 22 (2009).

20 TDDFT-TSH: Applications Protonated formaldimine TDDFT-TSH: Applications Protonated formaldimine Nonadiabatic couplings kj = Ṙ d kj States population PRL, 98, (2007); THEOCHEM, 914, 22 (2009). PRL, 98, (2007); THEOCHEM, 914, 22 (2009). TDDFT-TSH: Applications Protonated formaldimine States population - Average over many trajectories. Dashed line = CASSCF result. TDDFT-TSH: Applications Protonated formaldimine Geometrical modifications

21 TDDFT-TSH: Applications TDDFT-TSH: Applications Oxirane - Crossing between S 1 and S 0 Protonated formaldimine Comparison with experiment and model calculations In addition to the isomerization channel, intra-molecular proton transfer reactions was observed (formation of CH 3 NH + ). Oxirane Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation can occur. H 2 abstraction is also observed in some cases. Structures and life times are in good agreement with reference calculations performed using high level wavefunction based methods. Figure: Mechanism proposed by Gomer and Noyes Oxirane TDDFT-TSH: Applications Oxirane - Crossing between S 1 and S 0 Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation can occur. Computational details Isolated system Oxirane TDDFT-TSH: Applications Oxirane - Crossing between S 1 and S 0 Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation can occur. LR-TDDFT/PBE/TDA SH-AIMD 30 trajectories (NVT) each of 100 fs.

22 TDDFT-TSH: Applications Oxirane - Crossing between S 1 and S 0 The photophysics of solvated Ruthenium(II) tris-bipyridine [Ru(bpy) 3)] 2+ dye: photophysics [Ru(bpy) 3)] 2+ dye: Singlet state dynamics 1 Why Quantum Dynamics? N N y N z x N N N [Ru(bpy) 3)] 2+ dye: Solvent structure 2 Nonadiabatic Bohmian dynamics 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT [Ru(bpy) 3)] 2+ dye: triplet state dynamics 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S 1 and S 0 [M.E. Moret, I.T., U. Rothlisberger, JPC B, 113, 7737(2009); IT, B. Curchod, U. Rothlisberger, Chem.Phys., 391, 101(2011)] 5 Local control (LC) theory Addition of an external field within the equations of motion of TSH: Short summary of the theory The interaction Hamiltonian between the electrons and the td electric field is Ĥ int = e 2m ec A(r i, t) ˆp i i where A(r, t) isthe(classical)vectorpotentialoftheelectromagneticfield,ˆp i is the momentum operator of electron i, e is the electron charge, m e is the electron mass, and c is the speed of light. Startegy The idea is to induce electronic excitations through the direct interaction with the time-dependent (td) electric field instead of artificially promote the system into one of its excited states. Method: extendedtshnonadiabaticdynamics. Remark We are in the dipole approximation and therefore we do not need TDCDFT. IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81, (2010) IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81, (2010)

23 External field within TSH It can be shown (Phys. Rev. A (2010)) that through the coupling with the td electric field, Tully s propagation equations acquire an additional term with i~ C J (t) = I and where A 0 (t) =A 0 e CI (t)(h JI i~ṙ d JI + i! A 0 JI c µ JI e i!t ) i! JI A 0 (t) c µ JI = h J Ĥ int I i i!t is the vector potential of the external td electric field, µ JI = eh J i is the the transition dipole vector, and! JI =(E J E I )/~. ˆr i I i E ect of an electromagnetic field - Lithium fluoride Di erent excitations can be obtained, depending on the polarization vector of the laser pulse. Electronic structure of LiF Ground state - symmetry (GS). First excited state (doubly degenerate) - symmetry (S 1 ). Second excited state - symmetry (S 2 ). Avoided crossing between GS and S 2 Note that Tully s hops probability should be modified accordingly. IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81, (2010) Local control (LC) theory Local control theory Application: Photoexcitation of LiF in the bound state S 2 Control is achieved by tuning the temporal evolution of E(t) in a way to maximize the population of a target state. Using the TSH for the total molecular wavefunction 1 (r, R, t) = CJ (t) J(r; R) J for a given trajectory, thepopulationtimeevolutionsimplifiesto P I (t) = 2E (t) J =[C J µ JI CI (t)] It is now evident that choosing a field of the form E(t) = will ensure that P I (t) alwaysincreasesintime. T. J. Penfold, G. A. Worth, C. Meier, Phys. Chem. Chem. Phys. 12, 15616(2010). B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84, (2011) J = [CI (t)cj µ IJ )] Test system: LiF. The aim is to excite the molecule selectively into the S 2 bound-excited-state starting from the ground state. The dynamics is initiated in the ground state and the population is transferred from the GS into S 2 through direct coupling with an electron magnetic pulse (pi-piulse or local controlled pulse). B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84, (2011)

24 E ect of a generic polarized pulse LC pulse: e cient population transfer and stable excitation Comparison with wavepacket propagation (MCTDH) Local control on proton transfer (in gas phase) B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84, (2011)

25 Local control on proton transfer Local control on proton transfer Local control on proton transfer (with one water molecule) Local control on proton transfer (with one water molecule)

26 Local control on proton transfer (comparison) Local control on proton transfer (freq. vs. time) gas phase with one water molecule Local control on proton transfer (mode analysis) Recent review on TDDFT-based nonadiabatic dynamics ChemPhysChem,14, 1314(2013)