Ab initio nonadiabatic molecular dynamics TDDFT for ultrafast electronic dynamics Ivano Tavernelli and Basile Curchod LCBC EPFL, Lausanne TDDFT SCHOOL BENASQUE 2014 Ab initio nonadiabatic molecular dynamics Outline 1 Ab initio molecular dynamics Why Quantum Dynamics? 2 Mixed quantum-classical dynamics Ehrenfest dynamics Adiabatic Born-Oppenheimer dynamics Nonadiabatic Bohmian dynamics Trajectory Surface Hopping 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S1 and S0 5 TSH with external time-dependent fields Ab initio nonadiabatic molecular dynamics Outline Recent review on TDDFT-based nonadiabatic dynamics ChemPhysChem,14, 1314 (2013) Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics 1 Ab initio molecular dynamics Why Quantum Dynamics? 2 Mixed quantum-classical dynamics Ehrenfest dynamics Adiabatic Born-Oppenheimer dynamics Nonadiabatic Bohmian dynamics Trajectory Surface Hopping 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT 4 TDDFT-TSH: Applications Photodissociation of Oxirane Oxirane - Crossing between S1 and S0 5 TSH with external time-dependent fields Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Reminder from last lecture: potential energy surfaces We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei... Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Reminder from last lecture: potential energy surfaces We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei... Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Why Quantum dynamics? GS adiabatic dynamics (BO vs. CP) BO MI R¨I (t) = −∇ minρ EKS (fφi [ρ]g) ¨ δ δ Energy CP µi jφi (t)i = − EKS (fφi (r)g) + fconstr.g δhφi j δhφi j MI R¨I (t) = −∇EKS (fφi (t)g) ES nonadiabatic quantum dynamics Wavepacket dynamics (MCTDH) Trajectory-based approaches - Tully's trajectory surface hopping (TSH) - Bohmian dynamics (quantum hydrodyn.) - Semiclassical (WKB, DR) - Path integrals (Pechukas) Energy - Mean-field solution (Ehrenfest dynamics) Density matrix, Liouvillian approaches, ... Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Why Quantum dynamics? GS adiabatic dynamics First principles Heaven Energy 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 First principles Heaven Ab initio MD with WF methods Ab initio MD with DFT & TDDFT [CP] # Energy Models #? No principles World Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Why Quantum dynamics? GS adiabatic dynamics First principles Heaven Energy 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 (-) We cannot get read of electrons (-) Nuclei keep some QM flavor (-) Accuracy is an issue Energy (-) Size can be large (diffuse excitons) (+) Time scales are usually short (< ps) Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Nonadiabatic effects requires quantum nuclear dynamics The nuclear dynamics cannot be described by a single classical trajectory (like in the ground state -adiabatically separated- case) Ab initio nonadiabatic molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Why trajectory-based approaches? W1 In \conventional" nuclear wavepacket propagation potential energy surfaces are needed. W2 Difficulty 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 offer a good compromise between accuracy and computational effort. Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Starting point The starting point is the molecular time-dependent Schr¨odingerequation : @ H^ Ψ(r; R; t) = i Ψ(r; R; t) ~@t where H^ is the molecular time-independent Hamiltonian and Ψ(r; R; t) the total wavefunction (nuclear + electronic) of our system. In mixed quantum-classical dynamics the nuclear dynamics is described by a swarm of classical trajectories (taking a "partial" limit ~ ! 0 for the nuclear wf). In this lecture we will discuss two main approximate solutions based on the following Ans¨atzefor the total wavefucntion 1 Born- X Ψ(r; R; t) −−−! Φj (r; R)Ωj (R; t) Huang j Z t Ehrenfest i 0 0 Ψ(r; R; t) −−−−−−! Φ(r; t)Ω(R; t) exp Eel (t )dt ~ t0 Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Tarjectory-based quantum and mixed QM-CL solutions We can \derive" the following trajectory-based solutions: Nonadiabatic Ehrenfest dynamics dynamics I. Tavernelli et al., Mol. Phys., 103, 963981 (2005). Adiabatic Born-Oppenheimer MD equations Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT, U. Rothlisberger, PCCP, 13, 32313236 (2011) Nonadiabatic Trajectory Surface Hopping (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88, 166402 (2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95, 163001 (2005) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98, 023001 (2007) Time dependent potential energy surface approach based on the exact decomposition: Ψ(r; R; t) = Ω(R; t)Φ(r; t). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105, 123002 (2010) Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics Z t Ehrenfest i 0 0 Ψ(r; R; t) −−−−−−! Φ(r; t)Ω(R; t) exp Eel (t )dt ~ t0 Inserting this representation of the total wavefunction into the molecular td Schr¨odingerequation and multiplying from the left-hand side by Ω∗(R; t) and integrating over R we get 2 »Z – @Φ(r; t) X 2 ∗ i = − ~ r Φ(r; t) + dR Ω (R; t)V^ (r; R)Ω(R; t) Φ(r; t) ~ @t 2m i e i 2 e2Z where V^ (r; R) = P e − P γ . i<j jri −rj j γ;i jRγ −ri j In a similar way, multiplying by Φ∗(r; t) and integrating over r we obtain 2 »Z – @Ω(R; t) ~ X −1 2 ∗ ^ i~ = − Mγ rγ Ω(R; t) + dr Φ (r; t)Hel Φ(r; t) Ω(R; t) @t 2 γ Conservation of energy has also to be imposed through the condition that dhH^ i=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. Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation We start from the polar representation of the nuclear wavefunction i Ω(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 Ω(R; t) and separating the real and the imaginary parts one gets for the phase S in the classical limit ~ ! 0 Z @S 1 X 2 = − M−1r S − dr Φ∗(r; t)H^ (r; R)Φ(r; t) @t 2 γ γ el γ This has the form of the "Hamilton-Jacobi" (HJ) equation of classical mechanics, which establishes a relation between the partial differential equation for S(R; t) in configuration space and the trajectories of the corresponding (quantum) mechanical systems. Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation Z @S 1 X 2 = − M−1r S − dr Φ∗(r; t)H^ (r; R)Φ(r; t) @t 2 γ γ el γ Instead of solving the field equation for S(R; t), find the equation of motion for the corresponding trajectories (characteristics). Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation The identification of S(R; t) with the "classical" action, defines a point-particle dynamics with Hamiltonian, Hcl and momenta P = rRS(R): The solutions of this Hamiltonian system are curves (characteristics) in the (R; t)-space, which are extrema of the action S(R; t) for given initial conditions R(t0) and P(t0) = rRS(R)jR(t0). Newton-like equation for the nuclear trajectories corresponding to the HJ equation dP Z γ = −∇ dr Φ∗(r; t)H^ (r; R)Φ(r; t) dt γ el Ehrenfest dynamics @Φ(r; R; t) i = H^ (r; R)Φ(r; R; t) ~ @t el MI R¨I = −∇I hH^el (r; R)i Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation The identification of S(R; t) with the "classical" action, defines a point-particle dynamics with Hamiltonian, Hcl and momenta P = rRS(R): The solutions of this Hamiltonian system are curves (characteristics) in the (R; t)-space, which are extrema of the action S(R; t) for given initial conditions R(t0) and P(t0) = rRS(R)jR(t0). Newton-like equation for the nuclear trajectories corresponding to the HJ equation dP Z γ = −∇ dr Φ∗(r; t)H^ (r; R)Φ(r; t) dt γ el Ehrenfest dynamics - Densityfunctionalization (φk : KS orbitals) @ 1 2 i~ φk (r; t) = − rr φk (r; t) + veff[ρ, Φ0](r; t) φk (r; t) @t 2me MI R¨I = −∇I E[ρ(r; t)] Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - Example Ehrenfest dynamics @ 1 2 i~ φk (r; t) = − rr φk (r; t) + veff[ρ, Φ0](r; t) φk (r; t) @t 2me MI R¨I = −∇I hH^el (r; R)i h (no C4+) i ρ(r; t) − ρ0 (r) Ab initio nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics and mixing of electronic states Ehrenfest dynamics @Φ(r; R; t) i = H^ (r; R)Φ(r; R; t) ~ @t el MI R¨I = −∇I hH^el (r; R)i Consider the following expansion of Φ(r; R; t) in the static basis of electronic wavefucntions fΦk (r; R)g 1 X Φ(r; R; t) = ck (t)Φk (r; R) k=0 2 The time-dependency is now on the set of coefficients fck (t)g (jck (t)j is the population of state k).