Introduction Ehrenfest Surface Hopping Sharc Example Lecture 6: Introduction to Surface Hopping and to Sharc Sebastian Mai Sharc Workshop Vienna, October 3rd–7th, 2016 Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 1 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Introduction What we already have heard during the last days: I Classical nuclear dynamics I Ab initio dynamics I Born-Oppenheimer approximation I Excited states Goal: Perform excited-state dynamics with classical nuclei. We need to consider the Born-Oppenheimer approximation: I The electronic wavefunction can change during dynamics. I The electronic wavefunction has an eect on the nuclear motion. Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 3 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Electronic wavefunction evolution Classical nuclear dynamics No nuclear wavefunction! ) Electronic wavefunction as linear combination of basis functions: X Ψ(t) = c (t) ϕ (1) j i α j α i α Inserting this into the TDSE: @ i~ Ψ(t) = Hˆ el Ψ(t) (2) @t j i j i and premultiplying with ϕ gives: h β j * + X @c (t) D E @ X D E i~ α ϕ ϕ + c (t) ϕ ϕ = c (t) ϕ Hˆ el ϕ (3) @t β α α β @t α α β α 2 α 3 α 6 7 6 7 46 57 Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 4 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Electronic equation of motion From last slide: * + X @c (t) D E @ X D E i~ α ϕ ϕ + c (t) ϕ ϕ = c (t) ϕ Hˆ el ϕ (4) @t β α α β @t α α β α 2 α 3 α 6 7 6 7 Defining46 matrix elements: 57 X @c (t) X i~ α δ + c (t) T = c (t) H (5) @t βα α βα α βα 62 α 73 α 6 7 Rearranging:46 57 @cβ (t) X i = c (t) H + T (6) @t − α ~ βα βα α As matrix equation with atomic units: @ c(t) = [iH + T] c(t) (7) @t − Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 5 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Electronic wavefunction interpretation Complex electronic wavefunction represented through vector c(t). X Ψ(t) = c (t) ϕ (8) j i α j α i α Examples: ! 1 I c(t) = : wavefunction identical to the first basis function ϕ 0 j 1i ! p0:5 I c(t) = : wavefunction superposition of ϕ and ϕ p0:5 j 1i j 2i The description of the wavefunction depends on the basis functions, the representation! This will be a topic later. Before that: Ψ(t) must have eect on nuclear dynamics! j i Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 6 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Electronic wavefunction eect on nuclei First: Ψ(t) must have eect on nuclear dynamics! j i In quantum mechanics: I Wavepacket on ϕ follows gradient of ϕ j 1i j 1i I Wavepacket on ϕ follows gradient of ϕ j 2i j 2i : I : In classical mechanics: I Nuclei can only follow one gradient, cannot split! Two popular ways to get gradient: I Ehrenfest dynamics I Surface hopping Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 7 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Ehrenfest Dynamics Idea: Use the energy expectation value: D E X Ee = Ψ Hˆ el Ψ = c 2E (9) j α j α α Therefore also called mean-field dynamics. The corresponding gradient is: X X e 2 E = c E + c∗ c (E E )K (11) −∇R − j α j rR α α β β − α βα α α; β Electronic wavefunction coeicients show up in the gradient! Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 9 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Example of Ehrenfest dynamics 6 5 4 Energy (eV) 3 1 0 Coeicients 2 4 6 8 10 12 14 Coordinate (Å) Problems: I Single mean-field trajectory cannot split I Contributions from classically inaccessible states Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 10 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Problems of Ehrenfest dynamics 1. Several reaction paths I Single mean-field trajectory cannot split I Average path not representative for all reaction paths 2. Classically inaccessible states I Population on high-energy states increases mean energy I There should not be population in these states I Distortion of gradients Need a beer method than Ehrenfest… Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 11 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface Hopping Classical trajectories cannot split. Only solution for multiple reaction paths: Ensemble of many trajectories, plus statistics Central ideas of surface hopping I Many independent trajectories I Each trajectory moves in a pure state at any moment I Instantaneous hops between pure states I Hopping is stochastic, probabilities based on electronic wavefunction Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 13 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Example of Surface Hopping antum dynamics: 5 4 12 fs 3 2 Energy (eV) 1 0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Coordinate (Å) Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 14 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Example of Surface Hopping Single trajectory with surface hopping: 5 4 12 fs 3 2 Energy (eV) 1 0 2 | i c | 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Coordinate (Å) Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 14 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Example of Surface Hopping Ensemble with surface hopping: 5 4 12 fs 3 2 Energy (eV) 1 0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Coordinate (Å) I Final distribution of trajectories resembles distribution of wavepacket Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 14 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface Hopping Probabilities Eect of electronic wavefunction: decide which gradient to use. Electronic wavefunction: X Ψ(t) = c (t) ϕ (12) j i α j α i α Criterion for hopping probabilities: N (t) c (t) 2 = α (13) j α j Ntotal i.e., ensemble-averaged populations consistent with wavefunction. Also important: should make as few switches as possible! Too much switching leads to eectively averaging the energies like in Ehrenfest. Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 15 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface Hopping Probabilities Also important: should make as few switches as possible! Too much switching leads to eectively averaging the energies like in Ehrenfest. Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 16 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface Hopping Probability Equation Fewest-switching algorithm: 2 2 cβ (t) cβ (t + ∆t) hβ ? (t; t + ∆t) = j j − j j (14) ! c (t) 2 j β j Can be wrien as: 2 ∆t @ cβ (t) 2∆t @ j j = c∗ c (15) ≈ − c (t) 2 @t − c 2 < " β @t β # j β j j β j @ The term @t cβ is known from the equation of motion! We can calculate the hopping probabilities from the electronic wavefunction. Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 17 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Performing a Surface Hop At each time step: probability hβ α for each α. ! 0 Probability 1 Random number hβ 1 hβ 2 hβ 3 No hop → → → Get random number between 0 and 1 and find new state. Total energy conservation: I Adjustment of kinetic energy aer hop I If E E > E : No hop new − old kin I Can lead to inconsistency between classical population and quantum population Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 18 / 34 Introduction Ehrenfest Surface Hopping Sharc Example General Surface Hopping Algorithm t t + ∆t 6. Velocity-Verlet (v) R R v v Gact Gact β h c c H K Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 19 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface Hopping Advantages: I Classical: larger systems than quantum dynamics I Conceptually simple I On-the-fly ab initio is possible I Independent trajectories can be parallelized I Can describe several reaction paths Disadvantages: I Classical: Tunneling, Interference, Zero-point energy, … missing I Too much coherence Another problem: I Original surface hopping algorithm (Tully) only for internal conversion! Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 20 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Arbitrary couplings Would like to describe dierent processes: IC, ISC, laser excitation, … These are mediated by specific coupling terms in Hˆ el. Hˆ full = Hˆ MCH + Hˆ additional (16) I MCH: Molecular Coulomb Hamiltonian (only Ekin and Coulomb interaction, no relativistics/external fields) standard quantum chemistry ) I additional: relativistic eects (spin-orbit couplings), field-dipole interactions, … Spin-orbit couplings (SOC): I Relativistic eect: goes beyond Hˆ MCH I Coupling of intrinsic electron spin momentum with orbital angular momentum I Couples states of dierent multiplicity ISC ) Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 22 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Arbitrary couplings – Matrix representation of H^ MCH Matrix representation of Hˆ MCH: D MCH E Hβα = ϕβ Hˆ ϕα (17) With basis = diabatic states: HMCH = TMCH = * + * + . / . / . / . / , - , - With basis = eigenstates of Hˆ MCH: HMCH = TMCH = * + * + .