Real-space approaches for laser-molecule interactions Alejandro de la Calle, Abigail Wardlow and Daniel Dundas Atomistic Simulation Centre Queen’s University Belfast Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Talk Format • Motivation • Grid-based approaches • Solution of the TDSE for H + 2 – Quantum treatment of ionization and dissociation – Scaled cylindrical coordinates • Non-adiabatic quantum molecular dynamics for complex molecules – Time-dependent density functional theory – Adaptive real-space mesh techniques • Results • Outlook Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Motivation Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Motivation: Few Electron molecules • One electron (H 2 + ) and two electron (H 2 ) molecules • Solvable by theory – Study interplay between electron and dissociation dynamics – Correlated two-electron molecular dynamics – Understanding correlated electron-ion dynamics important in many areas Molecular electronics: Dundas et al, Nature Nanotech 4 99 (2009) • Easier to analyse in experiment – Fewer fragments – Analyse fragments simultaneously: distinguish dissociation from ionization Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Motivation: Many-electron molecules • Application in condensed matter physics, chemistry and life sciences • Elucidate the structure of biopolymers – Understand charge flow across the molecule Remacle & Levine, PNAS 103 6793 (2006) – Break specific bonds (molecular scissors) Laarmann et al, J Phys B 41 074005 (2008) • Control current flow in molecular electronic devices – Laser-controlled switching Kohler & H¨ anggi, Nature Nanotech 2 675 (2007) • Molecular identification – Enantiomer (chiral molecule) identification Lux et al, Angew Chem Int Ed 51 1 (2012) Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Grid-based Approaches Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Laser interaction with molecules: Physical processes involved 1. Multiphoton excitation and dissociation n ω L + AB → A + B 2. Multiphoton ionization n ω L + AB → AB + +e − 3. Dissociative ionization n ω L + AB → A a + + B b + +(a+b) e − 4. Raman scattering and high-order harmonic generation n ω L + AB → AB ∗ + m ′ ω ′ + m ′′ ω ′′ Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
General Approaches • For the range of molecules we want to describe we need to be able to deal with – Large regions of space – Long interaction times – Large data sets • We require parallel methods that scale to large numbers of processor cores – Sparse, iterative techniques – Retain high accuracy • Main class of methods considered – Adapted finite-difference grids – High-order explicit time propagators Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Finite difference techniques • Standard finite-difference technique: – Solve Schr¨ odinger equation on mesh of equally-spaced points – Approximate derivatives (Laplacian, etc) by central finite differences, e.g. � � d 2 − h 2 dx 2 f ( x ) = 1 12 f (4) ( η ) f ( x − h ) − 2 f ( x ) + f ( x + h ) h 2 where h is the step-size and x − h ≤ η ≤ x + h • Results in a highly sparse set of linear equations • Effective parallelization: nearest-neighbour communications (1 halo point) • Error ∝ h 2 – To reduce error: reduce h – In many cases error largest in small regions of space – Small step-size used in regions where not needed Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Adapted finite difference techniques • Can overcome these problems by using different coordinate scaling techniques – Global adaptation – Local adaptations • Scaling techniques with increasing grid spacing only valid for bound states – Equidistant grid spacing along direction of ionization • Need to be careful! – Resulting finite difference Hamiltonian is generally not Hermitian – Time propagation is not unitary – Effect is enhanced when very little ionization occurs • Can obtain Hermitian finite difference Hamiltonian – Derive Schr¨ odinger equation from appropriate Lagrangian Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Time propagation Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Time propagation techniques • In Krylov subspace methods – Calculate the vectors: Ψ , H Ψ , H 2 Ψ , ... , H N k Ψ – Orthonormalise these to form the vectors: q 0 , q 1 , q 2 , ... , q N k – Let Q be the matrix whose columns are the q ’s – h = Q † HQ is the Krylov subspace Hamiltonian • We propagate wavefunctions according to Ψ ( t + ∆ t ) ≈ e − iH ∆ t Ψ ( t ) ≈ Qe − ih ∆ t Q † Ψ ( t ) • Unitary to order of Krylov expansion E S Smyth et al, Comp Phys Comm 114 1 (1998) D Dundas, J Chem Phys 136 194303 (2012) Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Application of these methods Laser-molecule interactions Few-Electron Polyatomic Molecules molecules Numerical grid techniques Full-dimensional TDSE Quantum electrons (Electrons and Ions) Classical ions Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
General Approach for H + 2 Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Grid treatment of H + 2 • Light linearly polarized parallel to molecular axis • Full dimensional treatment of electron dynamics • 1-D treatment of nuclear dynamics H + r 1 e − 1 2 R r Laser O r 2 1 2 R H + Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Hamiltonian Hamiltonian for H + 2 can be written H tot ( R , r , t ) = T N ( R ) + H elec ( R , r , t ) H elec ( R , r , t ) = T e ( r ) + V ion ( R , r ) + U ( r , t ) • T N ( R ) : nuclear kinetic energy • H elec ( R , r , t ) : electronic Hamiltonian • T e ( r ) : electron kinetic energy • V ion ( R , r ) : Coulomb potential • U ( r , t ) : laser-electron interaction (length or velocity gauge) Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Time-Dependent Schr¨ odinger Equation (TDSE) We can derive the time-dependent Schr¨ odinger equation from the Lagrangian � � � � i ∂ d r Ψ ⋆ ( R , r , t ) L = d R ∂ t − H tot ( R , r , t ) Ψ ( R , r , t ) • Consider variation of Ψ ⋆ that leave action, A , stationary � t 1 δ A = δ L d t = 0 t 0 • Euler-Lagrange equation of motion � ∂ L ∂ L � ∂ Ψ ⋆ = d , ∂ ˙ Ψ ⋆ dt results in TDSE • Take variation after grid adaptation applied Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Coordinate scaling • Generalized cylindrical coordinates for electron dynamics r = g ( ρ ) cos φ i + g ( ρ ) sin φ j + h ( z ) k , • Laser linearly polarized along k direction, ⇒ no φ dependence • Volume element, d r = gg ′ h ′ d ρ d z = | J | d ρ d z • Electron kinetic energy � ∂ � ∂ � ∂ � gh ′ � gg ′ � ∂ρ + ∂ T e ( r ) = − 1 1 gg ′ h ′ g ′ h ′ 2 µ ∂ρ ∂ z ∂ z • Propagate the wavefunction = | J | − 1 / 2 ψ ( R , g , h , t ) � � R , g ( ρ ), h ( z ), t Ψ Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Coordinate scaling: Obtaining the TDSE • Lagrangian becomes � i ∂ � � � | J | d ρ d z | J | − 1 / 2 ψ ⋆ | J | − 1 / 2 ψ L = ∂ t − H tot ( R , r , t ) d R • Take variation with respect to ψ ⋆ gives TDSE ∂ 2 � � i ∂ψ − 1 ∂ R 2 − 1 T e − Z 1 − Z 2 ˜ ψ ∂ t = + U ( h , t ) 2 µ 2 M r 1 r 2 1 = g 2 + ( h − R / 2) 2 • r 2 2 = g 2 + ( h + R / 2) 2 • r 2 • M is reduced mass of the ions • µ is reduced mass of electron Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
Coordinate scaling: Obtaining the TDSE • Electron kinetic energy � ∂ � ∂ � g � 1 ∂ ∂ 1 1 1 1 ˜ √ gg ′ √ gg ′ + √ √ T e = g ′ h ′ ∂ρ ∂ρ ∂ z ∂ z h ′ h ′ = T ρ + T z • Symmetric expression when expressed in finite difference form • Can equally be applied to complex coordinate scaling • Simplify these to include second derivative terms – Reduces communications overhead in parallel simulations – D Dundas, J Chem Phys 136 194303 (2012) Quantum Dynamics In Systems With Many Coupled Degrees Of Freedom, Hamburg, Germany, 24–26 March 2014
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