Wave packet dynamics and Simulation of Pump-probe spectra Wave packet dynamics : Christoph Meier Universit´ePaul Sabatier, Toulouse, France time-dependent view of quantum mechanics Fundamental description of collision processes link with classical mechanics intuitive two aspects : wave packet dynamics as ’numerical tool’ for time-independent observables photoabsorption spectrum (first lecture) experimental realisation of wave packets: femtosecond spectroscopy (third lecture) Outline Lecture 1 Wave packet dynamics: General aspects Gaussian wave packets Wave packets as a link to classical mechanics Field-matter interaction: Time-dependent perturbation theory Photoabsorption spectra from a time-dependent view Lecture 2 Numerical methods of wave packet propagation Representation of wavefucntions Propagation methods A critical comparison Lecture 3 Femtosecond spectroscopy: Simulation of pump-probe spectra Lecture 1 Part 1 General aspects Gaussian wave packets connection with classical mechanics: Ehrenfest’s theorem Part 2 Field-matter interaction Time-dependent perturbation theory Photoabsorption spectra from a time-dependent view The 1-D Gaussian wave packet: V=0 the time evolution of a free Gaussian wave packet is given by: • p x = x + 0 t t 0 m pt = p0 α0 αt = 2ihα¯ 0t 1 + m expectation values x , p move classically ! • h i h i p (t = 0) x (t) = x (t = 0)+ h i t h i h i m p = p (t = 0) h i h i wave packet shows dispersion • 2 2 4¯h α0 v 2 ∆x = ∆x0 u1 + t u u m t 1 2 2 The 1-D Gaussian wave packet: V = 2mω x the time evolution of a Gaussian wave packet is given by: • p x = x cos(ωt) + 0 sin(ωt) t 0 mω pt = p0 cos(ωt) + mωx0 sin(ωt) α cos(ωt) + ia sin(ωt) α = a 0 t iα sin(ωt) + a cos(ωt) 0 mω here: a = 2¯h expectation values x , p move classically ! • h i h i width of wave packet changes periodically, except for α = a ! • 0 corresponds to the width of the ground state wavefunction = coherent state ⇒ this is an extremely good (fast) test for any numerical • propagation scheme ! Quantum-classical correspondence: Ehrenfest’s theorem Q: do the expectation values always move • according to classical mechanics ?: ∂χ(x,t) p2 ih¯ = + V (x) χ(x,t) ∂t 2m consider a time-independent observable A and a wavefunction χ: A = χ A χ dhAi h | | i h i = χ˙ A χ + χ A χ˙ dt h | | i h | | i 1 = χ [A, H] χ ih¯ h | | i specifically, for A = x and A = p we find: • d p x = h i (1) dth i m d ∂V p = (2) dth i −h ∂x i if ∂V ? ∂V = h ∂x i ∂x x= x h i holds, eq. (1,2) form a closed set of differential equations corresponding to Hamiltons equations in classical mechanics check: true for constant, linear, harmonic potential not true in general approximately true for a well-localized wavepacket: (hint: expand potential around center of wave packet, if harmonic approximation to potential holds across the width of wave packet = : OK ⇒ wave packet dynamics with Gaussians (E. Heller) • Lecture 1 Part 1 General aspects Gaussian wave packets connection with classical mechanics: Ehrenfest’s theorem Part 2 Field-matter interaction Time-dependent perturbation theory Photoabsorption spectra from a time-dependent view Maxwell’s equations and Electromagnetic Potentials electric and magnetic fields: 1 ∂B E = E =0 ∇× −c ∂t ∇· 1 ∂E B = B =0 ∇× c ∂t ∇· scalar and vector potential: 1 ∂A E = φ −∇ − c ∂t B = A ∇× A and φ obey: 2φ = 0 ∇ 1 d2A 2A = ∇ c2 dt2 hence a free-space solution is: A(r,t) = A sin(k r ωt) 0 · − which implies ω E(r,t) = A cos((k r ωt) εE cos((k r ωt) (3) c 0 · − ≡ 0 · − B(r,t) = A (k ε) cos((k r ωt) (4) 0 × · − electromeagnetic energy in volume V: 1 E d3r E2 B2 = V + 8π Z Energy density, photon flux and absorption spectrum electromagnetic energy in volume V: 1 E d3r E2 B2 = V + 8π Z using eq.(3,4) with E, B constant in volume V we obtain: E2 E = V 0 8π Einstein relation: energy of a single photon is quantized: hω¯ , hence E = Nhω¯ N relation between photon density V and classical electrical field amplitude E0: N E2 = 0 V 8πhω¯ photon flux: number of photons per unit area per unit time passing a particluar location: N E2c photon flux = c = 0 V 8πhω¯ absorption spectrum: transition rate σ(ω) = incident photon flux Field-Matter interaction Dipole approximation: • typical values for molecular spectroscopy in the optical range: laser wavelength: λ = 600nm dimensions of atom/molecule: 10 A˚ = fields are spacially constant at atomic level ⇒ = neglect k r electric and magnetic fields: ⇒ · A(r,t) A(t) ≡ E(r,t) E(t) ≡ B(r,t) B(t) ≡ Molecular field-free Hamiltonian with masses mj and charges qj: • 1 2 H0 = mjpj + V (r1, ) 2 Xj ··· Molecule-field interaction: minimal coupling, • qj pj pj A, identical to Lorenz force: −→ − c 1 qj 2 HC(t) = mj pj A(t) + V (r1, ) ! 2 Xj − c ··· Hamiltonian has become time-dependent by virtue of A(t) Field-Matter interaction 2 field-free Hamiltonian: • 1 2 H0 = mjpj + V (r1, ) 2 Xj ··· minimal coupling Hamiltonian HC: • 1 qj 2 HC(t) = mj pj A(t) + V (r1, ) ! 2 Xj − c ··· time-dependent Schr¨odinger equation: • ∂ψ (t) ih¯ C = H (t)ψ (t) ∂t C C change gauge by unitary transformantion: i qi ¯ j rj A(t) ψ(t) = e−h c · ψC(t) P leads to ∂ψ(t) ih¯ = (H (t) µ E(t)) ψ(t) ∂t 0 − · with µ being the dipole moment: µ = qjrj Xj note: no weak field assumption, only dipole approximation • Time-dependent perturbation theory for weak electric fields, the interaction energy W (t) = µ E(t) • − · can be viewed as time-dependent perturbation: ∂Ψ ih¯ = (H + W (t))Ψ(t) ∂t 0 change to interaction representation: • iH0t iH0t iH0t Ψ(t) = e Ψ(t) W (t) = e W (t)e− f g Schr¨odinger equation in interaction representation: ∂Ψ(t) ih¯ = W (t)Ψ(t) f∂t g f interaction starts at t0, one can integrate from t0 to t: i t Ψ(t) = Ψ(t ) dt′ W (t′)Ψ(t′) 0 t0 − h¯ Z f f g f iteration yields perturbative expansion: i t 1 t t′ Ψ(t) = Ψ(t ) dt′ W (t′)Ψ(t ) dt′ dt′′W (t′)W(t′′)Ψ(t ) + 0 t0 0 2 t0 t0 0 − h¯ Z − h¯ Z Z ··· f fzero g f g g f first second | {z } | {z } | {z } First order expression in Schr¨odinger representation: i i i H0(t t0) i t H0(t t ) H0(t t0) Ψ(t) = e−h¯ − Ψ(t ) dt′ e−h¯ − ′ W (t′)e−h¯ ′− Ψ(t ) 0 t0 0 − h¯ Z Electronic excitation by time-dependent perturbation theory First order expression in Schr¨odinger representation (t = 0): • 0 i i i H0t i t H0(t t ) H0t Ψ(t) = e−h¯ Ψ(0) dt′ e−h¯ − ′ W (t′)e−h¯ ′ Ψ(0) − h¯ Z0 suppose that the field induces a transition from an electronic • ground state g to an electronic excited state e : | i | i |e> |g> Born-Oppenheimer separation: • Ψ(t) = ψg(r, t) g > + ψe(r, t) e > | | nuclear wf nuclear wf | {z } | {z } if initially the molecule is in its electronic ground state g > | the light interaction leads to excitation = transfer of population to the excited state e >. ⇒ | Electronic excitation by time-dependent perturbation theory To calculate the excited state wavefunction, we project onto e : • | i t i i i h¯ He(t t′) h¯ Hgt′ ψe(r, t) = dt′ e− − µegE(t′)e− ψg(r, 0) −h¯ 0Z with Hg(r) = g H g = Tr + Vg(r) h | 0| i He(r) = e H e = Tr + Ve(r) h | 0| i µ = e µ g (r) eg h | | i this general expression is valid for arbitrary field shape: • = cw-absorption spectroscopy ⇒ = femtosecond spectroscopy ⇒ = control through pulse shaping ⇒ Absorption cross section from wave packet dynamics central relationship between absorption and dynamics • 2πω i ∞ ¯ (Eg hω¯ )t σ(ω) = dt φe(0) φe(t) e−h − 3¯hc Z h | i −∞ C(t) | {z } properties of the correlation function: • C(t) = C∗( t) − (= σ(ω) is real) ⇒ practical calculation: • 1. calculate ground state ψg ( wave packet dynamics) ←− 2. multiply ψg with transition dipole moment µeg: promoted state φe 3. propagate φe on excited state surface ( wave packet dynamics) ←− 4. calculate C(t) 5. take Fourier transform of C(t) key quantity: • C(t) = φe(0) φe(t) h | i φe(t) is a wave packet on the excited state surface • = exhibits time dependence ⇒ How to calculate φe(t) ? • wave packet propagation: • i ¯ Het φe(t) = e−h φe(0) Summary Part 1 General aspects Gaussian wave packets connection with classical mechanics: Ehrenfest’s theorem Part 2 Field-matter interaction Time-dependent perturbation theory Photoabsorption spectra from a time-dependent view Key references wave packets: general aspects, Gaussian wave packets: • C. Cohen-Tannoudji, B. Diu, F. Lalo¨e, Quantum mechanics, Vol 1 (Wiley, NY, 1977) E. J. Heller, J. Chem. Phys. 62, 1544 (1975), ibid. 65, 4979 (1976) spectra from correlation functions: • E. J. Heller, Acc. Chem. Res. 14, 368 (1981) E. J. Heller, S. Tomsovic, Physics Today, July 1993 R. Schinke, Photodissociation Dynamics, Cambridge University Press (1993) Lecture 2 Numerical methods of wave packet propagation • Representation of wavefunctions • grids basis pseudospectral methods DVR Propagation methods • Second Order Differencing Cayley’s method (Crank-Nicholson) Split-Operator A critical comparison (Chebyshev, Lanczos) Time-dependent self consistent field (TDSCF) Multi-Configuration Time-Dependent Hartree (MCTDH) Numerical methods for exact time propagation time-dependent Schr¨odinger equation: • ∂ ih¯ χ(t) = Hχ(t) ∂t i Ht χ(t) = e−h¯ χ(0) propagator | {z } problems: represent χ(0), χ(t) on computer • (= : approximation through truncation) ⇒ calculate/approximate propagator • apply propagator (or approx.