Lecture 6: Introduction to Surface Hopping and to Sharc

Introduction Ehrenfest Surface Hopping Sharc Example

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) | i α | α i α Inserting this into the TDSE: ∂ i~ Ψ(t) = Hˆ el Ψ(t) (2) ∂t | i | i and premultiplying with ϕ gives: h β | * + 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  α  α      

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  α  α     Defining matrix elements: 

X ∂c (t) X i~ α δ + c (t) T = c (t) H (5) ∂t βα α βα α βα  α  α   Rearranging:  ∂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) | i α | α i α Examples: ! 1 I c(t) = : wavefunction identical to the first basis function ϕ 0 | 1i ! √0.5 I c(t) = : wavefunction superposition of ϕ and ϕ √0.5 | 1i | 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! | 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! | i

In quantum mechanics: I Wavepacket on ϕ follows gradient of ϕ | 1i | 1i I Wavepacket on ϕ follows gradient of ϕ | 2i | 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) | α | 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 − | α | ∇R 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

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) | i α | α i α Criterion for hopping probabilities:

N (t) c (t) 2 = α (13) | α | 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) = | | − | | (14) → c (t) 2 | β | Can be wrien as:

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 Hˆ 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 = * + * + . / . / . / . / , Sebastian Mai- Lecture 6:Introduction, to Surface Hopping and- to Sharc 23 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Arbitrary couplings – Matrix representation of Hˆ full

Matrix representation of Hˆ full:

D full E Hβα = ϕβ Hˆ ϕα (18)

With basis = eigenstates of Hˆ MCH:

HMCH = TMCH = * + * + . / . / . / . / , - , - With basis = eigenstates of Hˆ full:

Hdiag = TMCH = * + * + . / . / . / . / , Sebastian Mai- Lecture 6:Introduction, to Surface Hopping and- to Sharc 24 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Representations of Hˆ MCH – Potentials

MCH Diabatic Eigenstates of Hˆ S2 E 1 1 E S ππ ∗ nπ ∗ 1 Coupling Coupling

Coordinate Coordinate

Choice of representation aects surface hopping dynamics: I Energetics I Localization of couplings: where/how oen to hop I Number of states necessary

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 25 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Representations of Hˆ full – Potentials

Eigenstate representations: full Diabatic states MCH eigenstates Hˆ eigenstates S2 2 3,4 5 3 nπ ∗ T1 E 1 1 E S E ππ ∗ nπ ∗ 1 1 Coupling Coupling Coupling

Coordinate Coordinate Coordinate Surface hopping done optimally in basis of eigenstates of Hˆ full: + All couplings localized less hops ⇒ + Multiplets treated correctly + Energetics most accurate Problem: antum chemistry programs calculate only eigenstates of Hˆ MCH but not of Hˆ full! Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 26 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Sharc

The basic idea of Sharc is to perform surface hopping in the diagonal representation, using only information from the MCH representation.

Transformation from MCH to diagonal representation:

diag MCH H = U†H U (19)

Can transform wavefunction:

diag MCH c = U†c (20)

Needs modifications to algorithms: I Propagating the electronic wavefunction using the MCH data I Calculating the hopping probabilities for the diagonal states I Geing the gradients of the diagonal states

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 27 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Electronic wavefunction propagation in Sharc

Na¨ıve way for propagation:

diag MCH MCH ∂ diag c (t + ∆t) = exp iU†H U + U†T U + U† U ∆t c (t) (22) − ∂t    * +  U†∂U/∂t is numerically diicult term.  ⇒  , - 

Numerically more stable propagation:

diag MCH MCH diag c (t + ∆t) = U(t + ∆t)† exp iU†H U + U†T U ∆t U(t) c (t) − f g (23) This is equivalent to three steps: diag MCH 1 Transform c (t) to c (t) MCH MCH 2 Propagate c (t) to c (t + ∆t) MCH diag 3 Transform c (t + ∆t) to c (t + ∆t)

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 28 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Surface hopping probabilities in Sharc

Propagation equation:

diag MCH MCH diag c (t + ∆t) = U(t + ∆t)† exp iU†H U + U†T U ∆t U(t) c (t) (24) | f− {z g } Pdiag (t+∆t,t)

Pdiag and cdiag are enough to calculate hopping probabilities:

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 29 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Gradients in Sharc

diag MCH Diagonal states are mixtures of MCH states (c = U†c ): X ϕdiag = U ϕMCH . (26) | β i βα | α i α

Diagonal gradients constructed similar to Ehrenfest gradient. I Needs gradients of MCH states I Needs nonadiabatic couplings between MCH states I Needs matrix U Linear combination of gradients and coupling vectors ⇒ Needs several quantities, so more expensive than regular surface hopping.

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 30 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Sharc in Action

I Wavefunction coeicients change suddenly (green and red curves)

I Induces hopping probability to red (S0 state) I Random number falls into range for hopping

I Trajectory switches to S0 state

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 32 / 34 Introduction Ehrenfest Surface Hopping Sharc Example Literature

J. C. Tully: Molecular dynamics with electronic transitions, J. Chem. Phys. 93, 1061 (1990). X. Li, J. C. Tully, H. B. Schlegel, M. J. Frisch: Ab initio Ehrenfest dynamics, J. Chem. Phys. 123, 084106 (2005). N. L. Doltsinis: Molecular dynamics beyond the Born-Oppenheimer approximation: Mixed quantum-classical approaches, in J. Grotendorst, S. Blugel,¨ D. Marx (editors), Computational Nanoscience: Do It Yourself!, volume 31 of NIC Series, 389–409, John von Neuman Institut for Computing, Julich¨ (2006). M. Barbai: Nonadiabatic dynamics with trajectory surface hopping method, WIREs Comput. Mol. Sci. 1, 620 (2011). S. Mai, P. Marquetand, L. Gonzalez:´ A general method to describe intersystem crossing dynamics in trajectory surface hopping, Int. J. antum Chem. 115, 1215 (2015).

Sebastian Mai Lecture 6:Introduction to Surface Hopping and to Sharc 33 / 34