PolyCEID: towards a better description of non-adiabatic molecular processes by Correlated Electron-Ion Dynamics Lorenzo Stella, R. Miranda, A.P. Horsfield, A.J. Fisher London Centre for Nanotechnology and University College London
PolyCEID Correlated Electron-Ion Dynamics (CEID) for Polymers (and molecules) Conjugated polymers Strong e-ion interaction The original CEID (see D. Dundas talk) not adequate New theory & new code Work in progress Source: www.electronic-paper.org.uk Source: Schwartz, Annu.Rev.Chem.Phys.
Key concepts Expanding quantum fluctuations of ions about their mean-field trajectory High order fluctuations needed for wave-packet splitting, especially for confined systems. Accuracy can be tuned, convergence to the exact Schrödinger dynamics Smooth dynamics, i.e., no jumps No need to diagonalise H e, good for metals Landau-Zener Crossing
Outline Introduction Exciton decay in conjugated polymers The PolyCEID algorithm Electronic correlations (R.Miranda) Conclusions and perspectives
Role of quantum fluctuation of ions EXCITON DECAY IN CONJUGATED POLYMERS
Electronic excitation in conjugated polymers Electron-ion interaction: the electronic structure is affected by the molecular conformation, and vice versa. Dimerised ground state Exciton-like excitation + - Polarons-like excitation (pair) - +
Non-radiative decay of excited conjugated polymers e.g., Polyacetylene The simplest semi-empirical model for π electrons: The Su-Schrieffer-Heeger (SSH) model: Lumped C-H atoms H t 2 0 ( 1 di, j )( ci c j h. c.) K di, j i, j 2 1 e-ion interaction (π electrons)
Role of quantum fluctuations of ions 32+2 atom SSH chain i Ri 1 2Ri 2Ri 1 Dim. ( ) 4 Chain dimerisation: deformation with respect to the relaxed GS 1-1 2-2 2-2 to 1-1 decay!
Non-radiative exciton decay Transition density matrix (R. Martin JCP 2003) ( h) ij k Tr ci ck ck c j0 Message: quantum fluctuations of ions can open new decay channels! 9
A new decay channel Mean-field (Ehrenfest) dynamics: Classical ions No quantum fluctuations No spontaneous electron decay CEID: Quantum ions Quantum fluctuations Electron decay by spontaneous emission of phonons
Theory and computation The PolyCEID ALGORITHM
Representation of the electron-ion state 2 We know the equations, but not the P i H e( R) solutions! Searching for the best t 2M approximation in a huge Hilbert space Mean-field trajectory CEID expansion order Matrix coefficients e-ion correlations CEID expansion: N CEID n, m n m Quantum fluctuations of the ions Ionic fluctuations: either Cartesian or normal modes CEID 0 Ehrenfest dynamics
CEID expansion to observables Expanding observables: O O n, m n, m n m Original CEID moments: O Tr O n, m n, mm, n Equal to 2 nd moment self-consistent CEID (D. Dundas) up to the 1 st order in the e-ion interaction. CEID EOMs n m 1 t i H, (corrections). n, m Derivatives of H here (1 st & 2 nd )
Sketch of the PolyCEID algorithm Input parsing Initial conditions Propagation Analysis
Sketch of the PolyCEID algorithm Input parsing Initial conditions (Still in progress) Flexible user interface. Possibility to customise the SSH Hamiltonian (different topologies, parametrisations, etc.) Propagation Analysis
Sketch of the PolyCEID algorithm Input parsing Initial conditions Propagation Analysis Factorised initial condition (Born-Oppeheimer), although a more general is possible as well I e Solve the adiabatic problem (electrons) H e( n n R) ( R) E ( R) ( R) e n n Steady solution (e.g., ground state) for the Ions 2 P 2M E n ( R), I n 0 PES Iterated to relax ions
Sketch of the PolyCEID algorithm Input parsing CI expansion (many determinats) Complete Active Space Initial conditions Propagation Analysis 1. Newton Eqs. (mean-field ions) 2. Ehrenfest Eq. (single-body e - ) 3. CEID Eqs. (many-body e - ) R P F P / M i H ( R) i Energy cut-off E det <E cut ij j Orbital basis set rotation (integrate fast quantum oscillations out)
Sketch of the PolyCEID algorithm Input parsing Initial conditions Propagation Analysis (Still in progress) Configurations must be saved (and reused) when it s possible. Observables and correlations functions computed at the end.
Study of convergence 2+2 site SSH model, resonant (Rabi oscillations)
Including Coulomb interaction ELECTRONIC CORRELATIONS
Multi-Configurational Time-Dependent Hartree-Fock (MTDHF) R. Miranda Pariser Parr Pople (PPP) Hamiltonian H H SSH U i 1 1 1 ( n )( n ) ( 1)( 1),, V,, i j i j n i i i n j 2 2 2 Level ordering of conjugated polymers (e.g., luminescence) CI expansion, coefficients fixed by symmetry (e.g., singlet), evolution of the single-particle orbitals, only! Dirac-Frenkel variational principle t 2 S dt H i t T ( R) t 1 i ij j
Importance of e-e interaction I: single chain Experimentally, triplet excitons are more localised Singlet Triplet
Importance of e-e interaction II: double chain E kin <E ee E kin E ee E kin >E ee Competition between: Kinetic term (hopping), delocalisation Potential term (Coulomb), localisation Three different regimes
Conclusions and perspectives Quantum fluctuations of ions can open new decay channels, spontaneous emission of phonons By CEID, quantum fluctuations can be included in a systematic and converging way Convergence demonstrated also for the electronic degrees of freedom, many-body states, CAS, and energy cut-off Electronic correlations by MCTDHF (R. Miranda) We want to merge the two approaches, complete description of nonadiabatic processes in molecular systems The project is growing, we re searching for new collaborators with expertises in quantum chemistry
Acknowledgments Andrew Fisher, LCN & UCL Andrew Horsfield, Imperial College Rafael Miranda, LCN & UCL EPSRC YOU!
PolyCEID: towards a better description of non-adiabatic molecular processes by Correlated Electron-Ion Dynamics THE END Lorenzo Stella, R. Miranda, A.P. Horsfield, A.J. Fisher London Centre for Nanotechnology and University College London
PolyCEID scaling Polynomial scaling The cost of a typical (i.e., nonoptimised) CEID simulation is: N modes N N modes CEID times more computational demanding than a typical Mean-field (Ehrenfest) simulation. 2 BUT, one can still reduce the cost by including only quantum fluctuations about relevant modes (physical insight)
Troubles with classical ions Electrons (fields): Infinite degrees of freedom Ions (particles): Finite degrees of freedom No thermalisation! e.g. Ehrenfest dynamics. No spontaneous emission (phonons) 28
Mean-field dynamics (Ehrenfest) Ions are heavy and (almost) localised Electrons are light and delocalised Mean-field forces into Newton s eq. (ions) F H ( t) ( R) ( t) R + time-dependent Schrödinger eq. (electrons) Several nice features: Insight from trajectories. No PES required (only Hamiltonian). Smooth dynamics i.e., no jumps. Good for metals and insulators 29
Adding quantum fluctuations: Correlated electron-ion dynamics (CEID) Density matrix electrons: ( t) ( t) ( t) TrI{ } e Ehrenfest dynamics ( R R) e Uncorrelated: CEID expansion: N CEID n, m n m Correlated CEID 0 Ehrenfest dynamics: 0,0 0 0 30
CEID equations of motion Earlier formulation: A.Horsfield et al., J.Phys.:Condens.Matter 17 (2005) t 1 H, i + Von Neumann equation: CEID n, m n m R) CEID expansion: ( R) ( t n. m CEID EOM: H, (corrections) N = 1 i Derivatives of H here (1 st & 2 nd ) n, m (Exact) 31
Theoretical model Hϋckel model π-electrons, hopping depends on bond length (SSH model) polarons, excitons No e-e interaction. 1D model, C-H groups are sites of a chain. 32 +2 sites, open boundary conditions (electrons). Initial conditions: dimerised chain, excited state. Fluctuations: normal modes expansion. Most coupled mode: dimerisation mode. Electronic configuration: 10 Slater s determinants. Lorenzo Stella --- CMS 2008 32
CEID equations of motion Lorenzo Stella --- CMS 2008 33
Many-Body space Single-particle orbitals: evolved by Ehrenfest dynamics Rotationg frame Theorem: Ehrenfest dynamics conserve the filling pattern Lorenzo Stella --- CMS 2008 34
Quantum coherence 2 Many-body: Tr e Tr 1 e Tr 2 Single particle: e e e Tr N fermionicity Idempotency: 2 e e Pure state Lorenzo Stella --- CMS 2008 35
MCTDHF theory Wavefunction ansatz jª i = X º c º j º i Dirac-Frenkel time-dependent variational principle µ h±ª j ^H i~ @ µ ª i + h ^H i~ @ ª j±ª i = 0 @t @t Result for a general open-shell state ^R = X ^P ¹ nº ^F º n ¹ ^F ¹ ^P º n º n ¹ ¹,º (n ¹ 6= n º ) ^F ¹ = ^T + 1 2 X X ³ 2a ¹ º ^J jº b ¹ º ^K jº n º º j º Lorenzo Stella --- CMS 2008
MCTDHF theory Consider a system of Ninteracting electrons, described by ^H = X i,j T ij ^c y i ^c j + 1 2 X i,j,k,l V ijkl ^c y i ^c y j ^c l ^c k The wavefunction is written as a superposition of Slater determinants with fixed coefficients jª i = X º c º j º i The goal is to find an optimal time evolution for the single-particle orbitals f Á i. gwe wish to approximate X i~j ª _ i = ^Hjª i ¼ ^Rjª i = R ij ^c y i ^c j jª i i,j Lorenzo Stella --- CMS 2008
MCTDHF theory In the Dirac-Frenkel formulation of the time-dependent variational principle, one varies the action integral S ª = Z t 2 hª j ^H i~ @ @t jª i dt t 1 with fixed end points This procedure yields µ h±ª j ^H i~ @ µ ª i + h @t ^H i~ @ @t ª j±ª i = 0 Lorenzo Stella --- CMS 2008
MCTDHF theory To write the variation, consider a small rotation of the set f Á i g já 0 j i = X i e ij já i i with ij = ji This gives j±ª i = X i,j ij ^c y i ^c j jª i Lorenzo Stella --- CMS 2008
MCTDHF theory The variational equation now reads X µ ij hª j ^H, ^c y X i ^c j jª i R kl hª j ^c y k ^c l, ^c y i ^c j jª i i,j k,l = 0 This methodology incorporates the orthonormality constraints by construction, and facilitates the elimination of redundant parameters Additionally, it can be shown that energy is conserved throughout the dynamics Lorenzo Stella --- CMS 2008
MCTDHF theory Application to an open-shell state yields ^R = X ^P ¹ nº ^F º n ¹ ^F ¹ ^P º n º n ¹ ¹,º (n ¹ 6= n º ) where the Fock operator for shell is ¹ ^F ¹ = ^T + 1 2 X X n º º j º ³ 2a ¹ º ^Jjº b ¹ º ^Kjº For an open-shell singlet 0 a = @ 1 1 1 1 0 1 1 1A b = @ 1 1 1 1 1 2 2 A 1 1 1 1 2 2 Lorenzo Stella --- CMS 2008