Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials. Oleg Prezhdo

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1 Nonadiabatic Molecular Dynamics with Kohn-Sham DFT: Modeling Nanoscale Materials Oleg Prezhdo CSCAMM Mar 11, 2010

2 Outline Nonadiabatic MD with Kohn-Sham DFT Advantages & Validity Quantum Backreaction & Branching (Nuclear Dynamics) Decoherence & Zero-Point Energy Carbon Nanotubes & Nanoribbons Luminescence Quenching Singlet-Triplet Transitions Structural Defects

3 Adiabatic vs. Nonadiabatic MD Nonadiabatic MD: Coupling between potential surfaces opens channels for system to change electronic states. transition allowed electrons treated quantum-mechanically e e e e e e nuclei treated classically weak coupling strong coupling

4 Time-Domain DFT for Nonadiabatic Molecular Dynamics ( ) = p p x x 2 ) ( ϕ ρ ( ) ( ) ( ) SD N v q p t x t x t x,, 2, 1 ϕ ϕ ϕ K = Ψ Electron density derives from Kohn-Sham orbitals H ( ) t R DFT functional depends on nuclear evolution ( ) ( ) t x H t t x i p p,, ϕ ϕ = h K = 1,2 p Variational principle gives = R i c c i R β α αβ β β β α χ χ δ ε h h Orbitals are expanded in adiabatic KS basis ( ) ( ) ( ) = α α α χ ϕ x t c t x p p, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t R x t R t R x t R x H ; ; ; α α α χ χ = ε

5 Time-Domain DFT in Many-Body Kohn-Sham Basis Craig, Duncan, Prezhdo PRL 95, (2005) non-zero only if different in one orbital

6 Open Theoretical Questions How to couple quantum and classical dynamics? quantum influence on classical trajectory Can one do better than classical mechanics for nuclear motion? zero-point motion, tunneling, branching, loss of coherence

7 Total energy of electrons and nuclei Nuclear Evolution: Ehrenfest Interfacial ET Stier, Prezhdo JPC B (2002) E M R tot = + V R t + Trx ρ x H x; 2 de tot is conserved 2 dt ( ()) ( ) ( R( t) ) = 0 time-dependent Hellmann-Feynman theorem gives Newton equation M R quantum force ( x) R H ( x R( t) ) = R V Tr ρ ; x

8 Nuclear Evolution: Surface Hopping a.k.a., quantum-master equation with time-dependent transition rates: - non-perturbative - correct short time dynamics Trajectory branching: Tully, JCP 93, 1061 (1990); Velocity Rescaling: Tully, Hammes-Schiffer JCP. 101, 4657 (1994). Detailed balance: Parahdekar, Tully JCP 122, (2005) Within TDDFT: Craig, Duncan, Prezhdo PRL 95, (2005)

9 Why Kohn-Sham Basis Works with Our Systems 1. KS excitations close to LR/TDDFT (in contrast to HF and CIS) 2. No bond-breaking, conformational changes, etc. 3. Many-electron systems, single excitation is a small perturbation 4. Averaging over many initial conditions and pathways Electron Transfer Example:

10 Silicon Quantum Dot KS and LR agree better for PW91 (pure DFT) than B3LYP (hybrid)

11 Small Molecule With Isomerization LR/TDDFT KS/TDDFT works well even in this case?! Tapavicza, Tavernelli, Rothlisberger PRL 98, (2007): LR/TDDFT

12 Schrodinger Cat and Decoherence B > 0 atom alive cat System - radioactive atom; Bath - cat dead In Nanomaterials System - electrons, spins; Bath - phonons

13 Franck-Condon Factor and Decoherence B > 0 ΣB 2 B 1 B 2 2 δ E1 E 2 = e ie 1 E 2 t/h B 1 t B 2 t dt Bath (vibrational) wave functions diverge This affects evolution of (electronic) system

14 Decoherence and Surface Hopping Reduced density matrix: ρ 11 ρ 12 ρ 21 ρ 22 ρ 11 ρ 12 B 2 B 1 ρ = B ρ S B B ρ 12 0 on decoherence time scale hopping probability P 12~ρ 12 ρ 21 B 1 B 2 ρ 22

15 Quantum Zeno Effect With decoherence: Without decoherence 2 P 12 = T 12 + T P 12 = T 12 + T Decoherence makes transitions less likely <

16 dψ = ih Stochastic Mean-Field (SMF) O. V. Prezhdo J. Chem. Phys. 111, 8366 (1999); Phys. Rev. Lett. 85, 4413 (2000) Stochastic Schrodinger equation in place of regular SE in Ehrenfest Ψ dt γ + L L Ψ dt+ 2 L system-bath interaction γ decoherence rate Advantages 1. Includes decoherence 2. Gives branching γ L Ψ 3. Infinitesimal velocity rescalings dw MF SH MF/SH SMF

17 Decoherence Induced Surface Hopping (DISH) Evolve in an adiabatic state. Hop when a decoherence event occurs. Rescale velocity as before in SH. Advantages DISH SMF 1. Includes decoherence 2. Gives branching 3. Nuclear evolution in pure states

18 Quantized Hamilton Dynamics O. V. Prezhdo, Y. V. Pereverzev J. Chem. Phys. 113, 6557 (2000) O. V. Prezhdo Theor. Chem. Acc. 116, 206 (2006) V = q2 2 + q3 3 d < q > dt = < p >; d < p > dt =- <q >- <q 2 > but < q 2 > < q > < q > and d < q 2 > dt = < pq + qp > 2 < pq > s d < pq > s dt = < p 2 >- <q 2 >- <q 3 > the infinite hierarchy is terminated by a closure < q 3 > 3 < q 2 > < q > 2 < q > 3

19 Harmonic Oscillator in Mapped QHD-2 hbar mass hbar mass QHD-2 takes care of zero-point-energy

20 Metastable Cubic Potential in Mapped QHD-2 QHD-2 gives tunneling hbar mass hbar mass

21 Quantum-Classical Lie Bracket O. V. Prezhdo, V. V. Kisil Phys. Rev. A (1997) O. V. Prezhdo J. Chem. Phys (2006) quantum commutator + classical Poisson bracket starting point for many methods: Ehrenfest, multiconfiguration quantum-classical problems with Jacobi identity: Alternative definition:

22 Bohmian Quantum-Classical Mechanics C. Brooksby, O. V. Prezhdo Phys. Rev. Lett. 86, 3215 (2001); 90, (2003) is ( r, t) / h ψ ( r, t) = ρ( r, t) e gives Newton eq. with non-local quantum potential 2 h Q( r) = 2m 2 r [ V ( r) Q( r) ] m&& r = r + ρ( r) ρ)( r) Q(r) Quantum (r) Classical (R) MR&& = R + m&& r = [ V ( R) V ( r, R) ] R rr [ V ( r) + V ( r, R) Q( r) ] r r rr + drop Q(R), Q( r, R) Advantage Branching

23 Example C. Brooksby, O. V. Prezhdo Phys. Rev. Lett. 86, 3215 (2001); 90, (2003) Highly simplified representation of O 2 interacting with Pt J. Strömquist, S. Gao, J.Chem.Phys. 106, 5751 (1997); D.S. Sholl, J.C. Tully, J.Chem.Phys. 109, 7702 (1998) H(q,Q) = T q q + T V qq Q Q V (q) = a(e + V (q) + V (Q) + V q V (Q) = 2 2 MΩ Q 2b(q c) (q,q) = Ae q 2 2e B(q Q) b(q c) ) qq (q,q) 1. Bohmian approach fixes asymptotic behavior 2. Bohmian and MF err at short times due to ZPE

24 Summary for Kohn-Sham SH KS-SH works well with nanoscale materials, even small molecules KS excitations are close to LR/TDDFT (in contrast to HF and CIS) Usually no bond-breaking, conformational changes, etc. Many-electron systems, single excitation is a small perturbation Averaging over many initial conditions and pathways FSSH gives (approx.) detailed balance, essential in applications Decoherence effects are essential in many-atom systems Stochastic Mean-Field (SMF) and Decoherence Induced SH (DISH) derive a SH algorithm from decoherence Zero-point-energy, tunneling and decoherence are included in Quantized Hamilton Dynamics (QHD) Bohmian quantum-classical formulation naturally gives branching

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27 Carbon Nanotubes Electronic Structure C h = na 1 + ma 2 Metallic (red dots) if n-m = 0 (mod 3) θ =0 Chiral θ =30

28 Electron-Phonon Relaxation and Phonon Modes High frequency optical G-modes Ma, Valkunas, Dexheimer, Bachilo, Fleming PRL 94, (2005) Low frequency acoustic radial breathing modes (RBM) * graphite background

29 Luminescence Quenching and Ground State Recovery Hertel et al. Nano Lett. 5, 511 (2005) Multiple luminescence quenching timescales in literature: ~20ps, 200ps and 10ns Luminescence yields are generally low <1%, i.e. quenching is efficient

30 Defects Stone-Wales defect bond rotation 7557 defect C 2 inserts across hexagon transition density

31 Defects and Density of States Ideal (6,4) tube bond rotation (Stone-Wales defect) Defect states both in the gap and inside bands C 2 inserts across hexagon (7557 defect) C 2 addition creates a distinct state E - E f (ev)

32 Decoherence Times & Fluorescence Linewidth Experiment 10 mev (suspended) 25 mev (surfactant) Defects create stronger electron-phonon coupling and can be detected by broader lines

33 Active Phonon Modes (8,4) n-m=1 mod3 (6,4) n-m=2 mod3 (7,0) n-m=1 mod3 SW 7557

34 Fluorescence Decay (FSSH) 40 ps 67 ps 150 ps Defects notably decrease lifetime, explain multiple decay components No T-dependence: cancellation of changes in NA coupling and decoherence

35 Comparison of FSSH with DISH FSSH DISH fixed coherence time FSSH and DISH give similar results DISH uses decoh. as SH algorithm Poisson and fixed coh. times similar DISH Poisson coherence time

36 Triplets & Phosphorescence Decay Triplets provide new decay pathway Jablonsky diagram Spin-Orbit (SO) coupling is on the same order as NA coupling, but SO coupling fluctuates less

37 Triplets & Phosphorescence Decay Triplets decay ~5 times more slowly than singlets Decay rate is proportional to square of SO coupling and inverse energy gap Larger tubes weaker coupling (due to smaller curvature), but also smaller energy gaps

38 Transition Densities Stone-Wales defect bond rotation 7557 defect C 2 inserts across hexagon

39 Phonon Modes disorder mode Opposite to tubes: weaker electron-phonon coupling in defects compared to ideal ribbon

40 Decoherence Time & Fluorescence Linewidth Opposite to tubes: wider lines in ideal ribbon than in defects

41 Fluorescence Decay (FSSH) 110 ps Defects speed up relaxation, same as in tubes Ribbons decay more slowly than tubes due to smaller electronic overlaps 300 ps

42 Comparison of FSSH with DISH FSSH DISH fixed coherence time DISH Poisson coherence time FSSH and DISH give similar results DISH uses decoh. as SH algorithm Poisson and fixed coh. times similar

43 Apparent Paradoxes Tubes vs. Ribbons Ideal ribbons show wider lines, i.e. faster dephasing, but slower relaxation than ideal tubes?? Ideal vs. Defects Defects accelerate relaxation in both tubes and ribbons; However, defects make lines broader in tubes and narrower in ribbons??

44 Summary for Carbon Tubes/Ribbons Luminescence quenching 3 timescales, as in experiment Tubes: 150ps ideal, 50ps defects, 700ps triplet channel Ribbons: 300ps ideal, 100ps defects Phonons: C-C stretch in ideal systems low frequency modes with defects No T-dependence by decoherence correction DISH and FSSH results are similar, DISH is more justified Phys. Rev. Lett (2006); Phys. Rev. Lett (2007); Nano Lett (2007); Phys. Rev. Lett., (2008); Nature Nanotech (2008); Pure & Appl. Chem (2008); Nano Lett (2008); Nano Lett (2008); J. Phys. Chem. C, 113, (2009); Nano Lett., 9, 12 (2009).