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Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Introduction to Quantum Dynamics: Solving the Time-Dependent Schrödinger Equation Graham Worth Dept. of Chemistry, University College London, U.K. 1 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Dynamical phenomena are described by the Time-Dependent Schrödinger Equation i~ ∂ Ψ(R, r, t) = ĤΨ(R, r, t) ∂t A wavepacket evolves in time driven by the Hamiltonian X i Ψ(q, t) = ci ψi e− ~ Ei t (1) (2) i where ψi are the eigenfunctions of the Hamiltonian • D.J. Tannor “Introduction to Quantum Mechanics: A Time-Dependent Perspective” (2007) University Science Books http://www.weizmann.ac.il/chemphys/tannor/Book/ • G. C. Schatz and M.A. Ratner “Quantum mechanics in chemistry” (2002) Dover • P.W. Atkins and R.S. Friedman “Molecular Quantum Mechanics” (2004) Oxford • K.C. Kulander “Time-dependent methods for quantum dynamics” (1991) Elsevier 2 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Aim of lectures: • Introduce Chemical Dynamics • Molecular Beams (scattering) • Time-resolved spectroscopy (femtochemistry) • The Time-dependent Schr"odinger Equation (TDSE) • Born-Oppenheimer Approximation. • Adiabatic and Diabatic Pictures • Techniques used to solve TDSE numerically • What is possible (bottlenecks / restrictions) 3 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Molecular Beams and Scattering Collimated beams of reactants intersect at right angles in high vacuum (> 10−7 Torr) Angular Distribution Crossed Molecular Beams Source B Velocity Distribution Source A Single collision (if any) occurs in crossing zone. 4 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Collisions may result in 3 types of scattering: • Elastic – Translational ∆E A + BC(ν, J) −→ A + BC(ν, J) • Inelastic – Rotational / vibrational ∆E A + BC(ν, J) −→ A + BC(ν 0 , J 0 ) • Reactive – New chemical products A + BC(ν, J) −→ AB(ν 0 , J 0 ) +C Must be able to distinguish new products from the background of elastic / inelastic scattered reactants. Implies sensitive and selective detector • Time-of-flight mass spectrometer (TOF) • “universal detector” • velocity and product identification • specific rotational / vibrational states probed by laser 5 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy The Cross-section b – impact parameter R, θ – coordinates Collision cross-section, σc , is effective target size. c Differential cross-section, dσ dω , is effective target size as a function Expect a minimum translation energy for reaction of scattering angle. Z 2π Z π dθ σc = 0 dφ 0 dσc dω Not every collision results in reaction Reaction cross-section σr < σc 6 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Example: F + D2 −→ DF + D Differential cross section at a relative energy of 1.82 kcal mol−1 shows probability of DF appearing at angle Θ and velocities (distance from scattering centre). • Contour map inhomogenous: Preferential orientations. • Mostly back scattered ⇒ head-on. • All collisions have same relative velocities (kinetic energies). Each reaction releases same energy, distributed between translational and internal (vib-rot) • Higher vibration ⇒ slower recoil Θ = 180◦ initial direction of F beam 7 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy F + H2 Potential Surfaces Product is hot with populated high vibrational states. Infrared chemiluminescence results – emission due to excited states generated in chemical reaction 8 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy H + H2 −→ H2 + H 1 Simplest “Reaction” Reaction Probability 0.8 2 ν=0→ν=0 0.6 ~ω = 0.27eV 0.4 0.2 ν=1 0 0.5 1.5 0.75 1 1.25 1.5 1.75 T =300K =0;1 [ A2] Energy [eV] 1 ν=0 0.5 0 0.5 1 1.5 Etrans 2 2.5 Reaction Probability 0.4 0.5 0.3 ν=1→ν=1 0.2 ~ω = 0.79eV 0.1 Reaction Cross-section (probability) for H + D2 0 0.8 1 1.2 1.4 1.6 1.8 2 Energy [eV] State-to-state cross-sections H + H2 9 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Pump-Probe Experiments: Femtochemistry 10 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Ultrafast molecular vibrations are the fundamental motions that characterize chemical bonding and determine molecular dynamics at the molecular level. Typical periods of motion: Vibrational ∼ 100 fs Rotational ∼ 100 ps (1 fs = 10−15 s) (1 ps = 10−12 s) Short (femtosecond) laser pulses allow us to “watch” the molecular motion Basic scheme: 1. pump laser pulse starts reaction 2. probe laser pulse probes molecules as reaction proceeds 3. Detection of probe signal 11 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Transient Spectra for NaI dissociation NaI∗ −→ [Na · · · I]‡∗ −→ Na + I Pump constant, change probe • (c) is resonant with Na D-lines • step-wise escape of Na • non-resonant same frequency • trapped portion of wavepacket • T = 1.2 ps 12 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Energetics described by the covalent (NaI) and ionic (Na+ I− ) potential energy curves which cross at an internuclear distance RC Non-adiabatic (2 interacting states). • In adiabatic picture curves do not cross • If system is adiabatic, bound-state • In diabatic picture curves cross • If system is diabatic, dissociation Which it is depends on coupling between states. 13 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Time-resolved study - Rhodopsin • Initial excitation - HOOP mode • after 50 fs S1 −→ S2 • energy −→ HT Kukura et al Science 310: 1006 (2005) 14 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy The Time-Dependent Schrödinger Equation i~ ∂ Ψ(R, r, t) = ĤΨ(R, r, t) ∂t (3) If the Hamiltonian is time-independent, formal solution Ψ(t) = exp −i Ĥt Ψ(0) Further, if we can write then (4) Phase factor Ψ(x, t) = Ψi (x)e−iωi t (5) '$ ∂ Ψ(x, t) = ~ωi Ψi (x)e−iωi t ∂t &% by comparison with the TDSE, Ψi are solutions to the time-independent Schrödinger equation i~ (6) ĤΨi = Ei Ψi = ~ωi Ψi (7) 15 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Ψi is a Stationary State as expectation values (properties) are time-independent hÔi = hΨi |Ô|Ψi ieiωi t e−iωi t = hΨi |Ô|Ψi i (8) If wavefunction is a superposition of stationary states, X χ(x, t) = ci Ψi (x)e−iωi t (9) i now, hÔi(t) = −i~ XX i ci∗ cj hΨi |O|Ψj iei(ωi −ωj )t (10) j An expectation value changes with time and depends on the initial function (ci coefficients). A non-stationary wavefunction is called a WAVEPACKET. 16 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Free Particle The functions E Ψk = eikx e−i ~ t represent a particle with an exact momentum p̂Ψk = −i~ d Ψk = k ~Ψk dx But, particle is not localised. Take a superposition Z ∞ χ(x, t) = dk C(k )Ψk (x, t) −∞ where C(k ) is a suitable function 17 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy E.g. Form a Gaussian wavepacket 2 −a (k − k0 )2 C(k ) = N exp 2 x − x0 (t)x0 (t) iγ χ(x, t) = N0 e exp − + ik0 x 2a2 δ where x0 (t) = ~k0 t m so wavepacket moves to right with velocity ~k0 m . The functions Ψk form a basis sutiable to describe free motion. 18 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Further, width of density, < x 2 > − < x >2 , is 2 2 ∆(t) = a (ln 2) 1 + ~ t m 2 a4 21 and as time increases. packet spreads out. k0 ~ t0 k0 ~ t0 + ∆t 19 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Bound Motion Ĥ = − E Ψ0 Ψ1 Ψ2 5 ~ω 2 3 ~ω 2 1 ~ω 2 ~2 ∂ 2 + 12 mω 2 x 2 2 2m ∂x 1 mω 2 2 = N0 e − 2 ~ x r mω 2 − 1 mω2 x 2 = N1 xe 2 ~ ~ 1 mω 2 2 mω 2 2 = N2 4 x − 2 e− 2 ~ x ~ The functions Ψk form a basis sutiable to describe bound motion. X χ(x, t) = ci (t)Ψi (x, t) i 20 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy The Born-Oppenheimer Approximation Start using Born representation X Ψ(q, r) = χi (q)Φi (r; q) , (11) i where electronic functions are solutions to clamped nucleus Hamiltonian Ĥel Φi (r; q) = Vi (R)Φi (r; q) . (12) The full Hamiltonian is Ĥ(q, r) = T̂n (q) + Ĥel (q, r) , (13) Integrate out electronic degrees of freedom to obtain 1 ∂χ − (∇1 + F)2 + V χ = i~ , 2M ∂t (14) 21 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy The Adiabatic Picture where Fij = hΦi | ∇Φj i (15) is the derivative coupling vector Assuming F M ≈0 ∂χ (16) ∂t and nuclei move over a single adiabatic potential energy surface, V , which can be obtained from quantum chemistry calculations. h i T̂n + V χ = i~ Unfortunately, Fij = hΦi | ∇Ĥel | Φj i Vj − Vi for i 6= j . (17) 22 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy The Diabatic Picture First we separate out a group of coupled states from the rest h i ∂χ(g) (g) (g) 2 (g) (T̂n 1 + F ) + V χ(g) = i~ ∂t , (18) To remove singularities, find a suitable unitary transformation Φ̃ = S(q)Φ (19) such that the Hamiltonian can be written [TN 1 + W] χ = i~ ∂χ ∂t (20) , where all elements of W are potential-like terms Worth and Cederbaum Ann. Rev. Phys. Chem. (2004) 55: 127 23 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy • Result 1: Electronic motion contained in potential energy surfaces which can be calculated using quantum chemistry • Problem 1: Potential surfaces are calculated in the adiabatic picture. Dynamics run in the diabatic picture Solution is to diabatise adiabatic surfaces for the dynamics. Non-trivial. 24 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Conical Intersections H Butatriene Radical Cation H C C C C H H CoIn 11 V [eV] 10.5 Diabatic FC 10 • 9.5 9 • • Xmin TS Amin • 8.5 • 11 90 -2 -1 0 -30 1 2 Q14 3 30 0 θ (deg) -60 4 -90 10.5 V [eV] 60 10 9.5 9 8.5 Adiabatic -90 -60 -2 -30 -1 0 0 1 Q14 2 30 3 4 θ 60 90 25 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Coordinates: The Kinetic Energy Operator In Cartesian coordinates, 3 1 X ∂2 T = − 2 2mi ∂xiα i=1 α=1 N X (21) This includes COM and ROT - continua. To remove these contributions use, e.g. Jacobi coordinates T C r θ Q Q Q Q Q Q B Q R QQQQ Q QQ Sukiasyan and JCP (02) : 116 A Meyer 1 ∂2 1 ∂2 = − − 2µR R 2 ∂R 2 2µr r 2 ∂r 2 1 1 +( + )j 2 2 2 2µR R 2µr r 1 − (J(J + 1) − 2K 2 ) 2 2µR R 1 p − (J(J + 1) − K (K ± 1)j± 2µR R 2 (22) 26 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy 6 Dimensional Jacobi Coordinates 2T̂ 3 X 1 1 1 ∂2 ~L†~L1 )BF = − R + ( + )( i 1 2 2 µi Ri ∂Ri2 µ R µ R 1 3 1 3 i=1 +( 1 1 + )(~L†2~L2 )BF 2 2 µ2 R2 µ3 R3 (~J 2 − 2~J(~L1 + ~L2 ) + 2~L1~L2 )BF + µ3 R32 . (23) Gatti et al JCP (05) 123: 174311 Other coordinates: Hyperspherical, Radau, .... 27 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Normal modes Final example, choose rectilinear coordinates so that force constant matrix (Hessian) is diagonal, Wij = ∂2V ∂xi ∂xj (24) then expanding around the minimum on the potential surface V = 3N−6 X i=1 ωi 2 Q + O(3) 2 i (25) COM and ROT removed and T = 3N−6 X i=1 ωi ∂ 2 − 2 ∂Qi2 (26) Very simple, but PES only suitable for small displacements. Wilson, Cross and Decius “Molecular Vibrations” (1980) Dover 28 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy • Result 2: Can select coordinates so that COM (and some ROT) motion removed and KEO has a simple form. • Problem 2: In general, simple KEO coordinates are not optimal for PES representation. In general, simple KEO coordinates are not optimal for PES representation and vice versa 29 / 30 Introduction Scattering Femtochemistry Potential Energy Surfaces Adiabatic / Diabatic Kinetic Energy Summary • Chemical physics is study of molecular interactions and resulting dynamics • Molecular beam scattering experiments provide details of interactions on ground-state • Cross-section relates to probability of process, e.g. reaction, occuring • Femtochemistry experiments probe dynamics on excited surface • pump-probe experiments create and watch wavepacket • Initialisation of a reaction creates a wavepacket, a solution of the TDSE • Starting point to solving the TDSE is the Born-Oppenheimer Approximation • Nuclear / electronic coupling leads to breakdown of BO • Adiabatic and Diabatic Pictures • To solve TDSE need Ĥ: PES + KEO 30 / 30