Non-adiabatic dynamics with external laser fields

Transcription

1 January 18, 21

2 1 MQCD Mixed Quantum-Classical Dynamics Quantum Dynamics with Trajectories 2 Radiation Field Interaction Hamiltonian 3 NAC NAC in DFT 4 NAC computation Sternheimer Implementation Nearly equilateral H 3 Protonated formaldimine, CH 2NH Appl. H + 2 H Appl. LiF LiF 7 Appl. Oxirane Oxirane

3 Mixed Quantum-Classical Dynamics Definitions {r, m} fast (light) degrees of freedom {R,M} slow (heavy) degrees of freedom Mean-field (Ehrenfest-type) dynamics R t t dt E el (t ) Ψ(r, R,t) = Φ(r,t)Ω(R,t)e i Z Z E el (t) = drdr Φ (r,t)ω (R,t)H el Φ(r,t)Ω(R,t) Born-Oppenheimer (BO) and non-adiabatic BO Ψ(r,R, t) = X I Φ I (r,r)ω I (R,t) Φ I (r,r) Solution of the time-independent Schrödinger equation (Only parametrically dependent on R) Other possible Ansätze

4 Quantum Dynamics with Trajectories Molecular Hamiltonian H tot = X γ z } { 2 X 2 2 γ 2 i + V en (r,r) 2M γ 2m i i H Inserting Ψ(r, R,t) = P I Φ I(r,R)Ω I (R,t) in the TDSE Nuclei Ω I (R, t) = A I (R,t)e i S I (R,t) in the limit ( ) ( A J = F(A t J, A J,S J, S J ) + P I J A J S J = P 1 t γ 2M γ ( γs J ) 2 + H II i hṙ d JI + i H IJ e i R t t dτ (H II H JJ ) Motion evolves on each PES with transport of amplitude between PESs governed by the off-diagonal terms d JI and H JI. Electrons are NOT dynamical. Define the PES computed on-the-fly

5 Quantum Dynamics with Trajectories Independent Trajectory Approximation In the ITA each classical trajectory representing the nuclear wavepacket evolve independently from all the other (missing coherence/decoherence effects) and the amplitudes only change through the coupling terms between PESs A J t = X I J A J»Ṙ d JI + i H IJ e i R t t dτ (H II H JJ ) J. Tully 1 has introduced a stochastic algorithm that (in the limit ) decide about the surface that drags the "classical" nuclear dynamics. 1 J.C. Tully, R.K. Preston, JCP, 55, 562 (1971)

6 Quantum Dynamics with Trajectories Tully s Surface Hopping In Tully s dynamics, the classical trajectories evolve adiabatically according to Born-Oppenheimer dynamics until a hop between two potential energy surfaces (H II and H JJ ) occurs according to a Monte Carlo-type procedure. In the "fewest switches" algorithm, the transition probability from state I to state J in the time interval [t, t + dt] is Z t+dt g IJ (t,t + dt) 2 dτ Im[A J(τ)A I H JI (τ)] Re[A J (τ)a I (τ)ξ JI (τ)] t A I (τ)a I (τ), (1) where Ξ JI (τ) = P γ Ṙγ d γ JI (τ) = P γ and only if X R γ K i 1g JK < ζ < X K I where ζ is generated randomly in the interval [, 1] Φ t I R γ Φ J, and a hop occurs if g JK, (2)

7 Quantum Dynamics with Trajectories Tully s Surface Hopping In the ITA a single trajectory follows adiabatically a PES (I) until a transition to a different PES (J) is accepted with a given probability g(a IJ ). The amplidudes A I evolve according to i Ȧ J = X I A I (H JI i Ṙ d JI ) A swarm of independent trajectories will then reproduce the approximate dynamics of the nuclear wavefunction amplitude. Tully s claim The algorithm "guarantee" that at any instant of time for the ENSEMBLE of trajectories the fraction of trajectories assigned to any PES is approximately equal to the relative population of the state A I (t) 2.

8 Quantum Dynamics with Trajectories Protonated formaldimine is a model compound for the study of isomerization in rhodopsin chromophore retinal. Photo-excitation promotes the system mainly into S 2 and therefore the relaxation occurring in the excited states involves at least 3 states: S (GS), S 1 and S 2.

9 Interaction Hamiltonian Interaction Hamiltonian In the Born-Oppenheimer approximation for the separation of electronic and nuclear degrees of freedom (which is assumed in Tully s dynamics), the total (non relativistic) Hamiltonian is given by H tot = H mol + H rad + H int where the interaction Hamiltonian (with no spin-magnetic field contributions) is obtained from the standard prescription p p ea/c, H int = X i» e 2mc (p i A(r i,t) + A(r i,t) p i ) + e2 2mc A(r i,t)a(r 2 i,t). The vector potential is of the form A = A ǫ λ e ik r+iωt

10 Interaction Hamiltonian In a typical molecular transition in the optical region, the wavelength of the interacting photon is much greater than the linear dimension of the molecule, λ = 2ω/k >> r molecule and therefore e ik r = 1 ik r + O(k r) 2 can be replaced by the leading term, 1. The radiation field coupling matrix element becomes (using A = A ǫ λ ) H int JI = iω JI A c µ JIe iωt where µ JI = e Φ J X α r α Φ I is the the transition dipole vector, and ω JI = (E J E I )/.

11 Interaction Hamiltonian Tully s differential equations for the state amplitudes i ȦJ = X I A I (H JI i Ṙ d JI) become i ȦJ = X I A i (H JI i Ṙ d JI + iω JI A c µ JI eiωt ) In addition, a classical electrostatic interaction term of the form E nucl (R α ) = X γ Z γr γ E(t) with E(t) = 1 A(t) is added to the potential energy of the nuclei, which, c t according to Tully s scheme, follow a "classical" trajectory on a single PES with possible surface hops according to the hopping probability g JI.

12 NAC in DFT NAC evaluation For any one-body operator, Ô, a mapping between MBPT and TDDFT quantities gives (we only consider transitions from the ground state Φ ) O S 1/2 e I = ω 1/2 I Φ Ô Φ I where the operator Ô = P iaσ o iaσâ aσâ iσ has components o iaσ = φ iσ Ô ψ aσ 2 with ω I = E I E and S ijσ,klτ = δ ik δ jl δ στ (f kσ f lσ )(ǫ lσ ǫ kσ ), e I are the TDDFT eigenvectors of the pseudoeigenvalue equation, Ωe I = ω 2 Ie I, q Ω ijσ,klτ = δ στ δ ik δ jl (ǫ lτ ǫ kσ ) q(f iσ f jσ )(ǫ jσ ǫ iσ )K ijσ,klτ (f kτ f lτ )(ǫ lτ ǫ kτ ) K is P the matrix form of the TDDFT kernel 2 iaσ stands for P N P σ {α,β}. P i=1 a=1

13 NAC in DFT The operator Ô can be the position operator, ˆr or the gradient R Ĥ ee, or any other one-body operator of interest, and may depend parametrically on the atomic positions, R. 3 For practical purposes we introduce the auxiliary linear-response many-electron wavefunctions 4 as a linear combination of singly excited Slater determinants Φ I [{φ }] = X iaσ c I iaσ â aσâ iσ Φ[{φ }], with c I iaσ s S 1 iaσ ω I e I iaσ where Φ [{φ }] is the Slater determinant of all occupied KS orbitals {φ iσ } N i=1, which, at a turn, are promoted into a virtual (unoccupied) orbitals, ψ aσ. 3 E. Tapavicza, IT, U. Rothlisberger, PRL, 98, 231 (27); C.P. Hu, O. Sugino, JCP, 127, 6413 (27) 4 IT, E. Tapavicza, U. Rothlisberger, JCP, 13, (27); JCP, 131, (29).

14 NAC in DFT The non-adiabatic coupling elements at the mid step t + δt/2 of a TDDFT MD dynamics can therefore be calculated as E Ṙ d I t+δt/2 = D Φ(r;R(t)) R ΦI (r; R(t)) D Φ(r; Ṙ = R(t)) t Φ E I (r;r(t)) 1 h i Φ (r;r(t)) Φ I (r; R(t + δt)) Φ (r; R(t + δt)) Φ I (r;r(t)) 2δt The non-adiabatic coupling vectors Ω I = Φ I (R) R Φ (R) = Φ I (R) R H Φ (R) E I (R) E (R) In second quantization, ˆr = P ia r iaâ aσâ iσ, and r is the vector with elements r ia = φ a ˆr ψ a in the basis of the KS orbitals. The transition dipole matrix elements µ I = eω 1/2 I r S 1/2 e I

15 Sternheimer Implementation Linear response perturbative TDDFT equations (Sternheimer 1951) NX j=1 (H σ δ ij ǫ ij ) φ I,jσ + Q σ δv σscf I φ iσ = ω I φ I,iσ where Q σ = 1 P N i=1 φ iσ φ iσ, H σ is the Kohn-Sham Hamiltonian, ǫ ij is a Lagrangian multiplier ensuring the orthogonality of the ground state orbitals, Z δv σscf I (r, ω) = d 3 r 1 r r + δ 2 E xc δρ σ(r) δρ σ(r )!ρ ρ=ρ Iσ (r, ω) (using ALDA) with ρ Iσ (r, t) = P i φ iσ(r)φ I,iσ (r, t) + φ I,iσ (r, t)φ iσ (r). In this formulation: Φ I [{φ }, {φ I, }] = X iσ ˆr I,iσâiσ Φ [{φ }] where ˆr I,iσ is the creator operator for the linear response orbital φ I,iσ.

16 Implementation The gradient of the Hamiltonian operator in Ω IJ = Φ I (R) R Φ J (R) = Φ I (R) R Ĥ Φ J (R) E I (R) E J (R) is split into the explicit and the implicit terms, R Ĥ[ρ](R) = Ĥ[ρ](R) Z ρ + R dr Ĥ[ρ](R) ρ(r) ρ(r) R, The first one can be computed analytically and describes the contribution from the electrostatic interaction between nuclei and electrons, and from the local and nonlocal parts of the pseudopotentials. The second term instead adds the contribution from the Hartree and exchange-correlation potentials.

17 Implementation There are two possible solutions of the implicit term: Using finite differences. Costly because there are 3 M nuclear displacements (M is the number of atoms). Using Density Functional Perturbation Theory (DFPT) with perturbing functional E pert γ [ρ] = Z drρ(r) V ions,el(r) R γ where the perturbation is described by an infinitesimal nuclear displacement, λ e Aµ, along the cartesian axis centered on each atom. The index A labels the ions and µ the cartesian coordinates along which the displacements occur (µ = 1, 2,3). We use the collective index γ for both indices A and µ. δρ γ(r) = X iσ φ iσ (r)δφ γ,iσ(r) + δφ γ,iσ (r)φ iσ(r)

18 Implementation The linear response-density is then used to compute the matrix elements of the implicit term of R Ĥ[ρ](R) according to Φ I X σ Z dr Ĥ[ρ](R) ρ σ(r) ρ σ(r) R Φ J = Φ I X γ X Z e γ σ dr Ĥ[ρ](R) ρ σ(r) where Φ J is either the ground state wavefunction (J = ) or 5 Φ J = NX X ˆr J,iσâJ,iσ Φ i=1 σ ρ γσ(r) Φ J 5 IT, E. Tapavicza, U. Rothlisberger, JCP, 13, 1241 (29)

19 Nearly equilateral H 3 Energy [ev] Ω 1 component [Bohr -1 ] R [Å] d12 x H 2 H 1 d 3 Upper panel: Adiabatic potential energy surfaces of the ground state and first singlet excited state of H 3 calculated along the path depicted here above. Solid lines: TDDFT/ALDA; Dashed lines: reference. lower panel: Corresponding z-component of the NACVs on the approaching H atom. The ALDA/TDDFT results (circles) are shown together with some reference data. Un-rescaled TDDFT/ALDA z components of the NACVs are also shown (crosses). a a IT, E. Tapavicza, U. Rothlisberger, JCP, 13, 1241 (29); THEOCHEM, 914, 22 (29). χ H 3 z

20 Protonated formaldimine, CH 2 NH Energy [ev] 1 5 Ω Ω 1 [Bohr [Bohr -1 ] ] Time evolution of the potential energy surfaces (upper panel) S (solid line), S 1 (dashed line), S 2 (dash dotted line), and S 3 (dotted line). The state that drives the dynamics is shown in red. The main structural change in S 2 during the first 1 fs is a CN bond elongation. The middle and lower panel show the corresponding time series of the absolute value of the NACVs of the carbon (black) and nitrogen (red) atoms computed for the pair of states S /S 1 and S 1/S 2, respectively fs

21 H + 2 Photoinduced excitation of H A.J. Cohen, P. Mori-Sanchez, W. Yang, Science, 321, 792 (28)

22 H H 2 + Rabi oscillations.8.6 C i fs

23 H + 2 H 2 + DFT BLYP Energy [a.u.] c 2 c r [Å] fs

24 H + 2 H 2 + DFT BLYP Energy [a.u.] c 2 c fs fs

25 H + 2 H 2 + HF Energy [a.u.] C 2 C r [Å] fs

26 H + 2 H 2 + HF Energy [a.u.] C 2 C fs fs

27 LiF Photoinduced excitation of LiF

28 LiF 1 LiF (1,,) - Without hops.8 C(t) GS S1 S2 S Time [fs]

29 LiF 1 LiF Pulse (1,,) - High intensity C(t)^ Li-F [Angstrom] Time [fs]

30 LiF C(t) 2 Li-F [Å] S1 S2 S3 GS LiF Pulse (1,1,1) A(t) Time [fs]

31 LiF LiF Pulse (1,1,1) Energy [hartree] Time [fs]

32 LiF 1 LiF Pusle (1,1,1) - High intensity - Out of phase (omega =.5 a.u.).8 C(t) Li-F [Å] Time [fs]

33 Oxirane Photoinduced excitation of Oxirane

34 Oxirane

35 Oxirane xxx

36 Oxirane Energy [hartree] C(t) S1 S2 S3 S4 GS Oxirane Succession of short pulses A(t) Time [fs]

37 Oxirane Acknowledgments Dr. Enrico Tapavicza (University of California Irvine) Basile Curchod (EPFL) Ursula Roethlisberger (EPFL) Also thanks to N. Maitra for introducing me to the photo-dissociation H + 2 system.