Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p.1
Outline Phenomena in laser-matter interaction: molecules Molecules, Born-Oppenheimer approximation Bond softening, enhanced ionization, Coulomb explosion High-harmonic generation in molecules p.2
Molecules Set of nuclear and electronic coordinates R j and r k. Hamiltonian: H 0 = Pj 2 2M j + p 2 k 2 + w j,e (R j,r k )+ j k j,k w j1,j 2 (R j1,r j2 ) + w ee (r k1,r k2 ) j 1 j 2 k 1 k 2 with w j1,j 2 nucleus-nucleus interaction, w j,e nucleus-electron interaction, and w ee electron-electron interaction. Light-molecule interaction: H(t) = H 0 D E(t) with dipole moment D = ( j Z jr j ) ( k r k) p.3
Born-Oppenheimer approximation Idea: separation of time scales for nuclear and electronic motion due to great mass difference Electrons adjust instantaneously to nuclear positions. Born-Oppenheimer (BO) Ansatz for wave function: Ψ(R,r,t) = m χ m(r,t) Φ m (R,r), Φ m (R,r) = electronic eigenstates at fixed nuclear positions. Inserting into the field-free TDSE yields i t χ m(r,t) = [ T n + V BO m (R) ] χ m BO approximation + m Φ m T n Φ m χ m nonadiabatic couplings (T n acting on both Φ m and χ m ) p.4
Born-Oppenheimer approximation Including the laser-molecule interaction, the BO TDSE becomes: i t χ m(r,t) = [ T n + V BO m (R) ] χ m (R,t) E m Φ m D Φ m χ m Functions χ m coupled only by the dipole matrix elements. BO approximation breaks down for highly excited electrons: Rydberg molecules Electrons in the continuum p.5
Fragmentation mechanisms in H + 2 Bond softening Above-threshold dissociation Charge resonance enhanced ionization Coulomb explosion p.6
Bond softening Transitions between BO states occur at nuclear positions where photon energy is resonant. 0.1 H 2 + V(R) (a.u.) 0.2 0.3 0.4 0.5 σ u ω 0.6 σ g 0.7 0 4 8 12 R (a.u.) p.7
Bond softening Alternative picture: potential surfaces shifted by multiples of the photon energy ( diabatic potentials) 0.1 H 2 + 0.2 V(R) (a.u.) 0.3 0.4 σ u ω 0.5 0.6 σ g 0.7 0 4 8 12 R (a.u.) Lowering of dissociation threshold = bond softening p.8
"Above-threshold" dissociation 3-photon absorption + 1-photon emission = effective 2-photon absorption 0.1 0.2 H 2 + V(R) (a.u.) 0.3 0.4 0.5 σ u 3ω 0.6 0.7 0.8 σ g 2ω 0 4 8 12 R (a.u.) ω p.9
Charge-resonance enhanced ionization At a range of internuclear distances, the tunneling barrier is suppressed by the presence of the second center. enhancement of ionization 1 V 0 (x) xe H 2 +, R=10 a.u. 0 V(x) (a.u.) 1 2 Seideman, Ivanov, Corkum, PRL 75, 2819 (1995), Zuo, Bandrauk, PRA 52, R 2511 (1995). 3 12 7 2 3 8 x (a.u.) p.10
Coulomb explosion Ionization of H + 2 and other molecular ions can create two charged centers rapid fragmentation due to Coulomb repulsion. + + Kinetic energy release indicates initial internuclear distance by energy conservation: E kin 1/R initial Coulomb explosion imaging p.11
Typical fragment spectrum Energies of D + ions from D 2 in a strong pulse Kinetic energy (ev) [Trump et al. PRA 62, 063402 (2000)] p.12
Harmonic generation in molecules For small molecules: electron excursion >> molecular size electron Atom-like mechanism photon Influence of molecular properties on ionization and recombination Probing of molecular structure / dynamics p.13
Two-center interference Recolliding electron with wave vector k in H 2 or H + 2 λ k k λ/2 Intensity (arb. u.) 10 2 10 4 10 6 θ=0 ο constructive interference destructive interference 10 8 0 1 2 3 4 Photon energy (a.u.) Minimum occurs when R cosθ = λ/2 with λ = 2π/k = electron wavelength [Fixed-nuclei TDSE calculations: M.L., N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, PRL 88, 183903 (2002) p.14
Multielectron dynamics in HHG Not only the highest occupied molecular orbital (HOMO), but also lower-lying orbitals participate in harmonic generation when HOMO contribution is suppressed by symmetry. Example: CO 2 O. Smirnova et al., Nature 460, 972 (2009) p.15
Strong-field approximation for high-harmonic generation (also known as Lewenstein model) D(t) = i t 0 dt E(t ) d 3 p p+a(t ) x 0 0 r p+a(t) exp( is) + c.c. where S(p, t, t t [ ] ) = (p+a(t )) 2 2 + I p t dt (length-gauge form) [Lewenstein et al., Phys. Rev. A 49, 2117 (1994)] p.16
Strong-field approximation for high-harmonic generation (also known as Lewenstein model) D(t) = i t 0 dt E(t ) d 3 p p+a(t ) x 0 0 r p+a(t) exp( is) + c.c. where S(p, t, t t [ ] ) = (p+a(t )) 2 2 + I p t dt (length-gauge form) [Lewenstein et al., Phys. Rev. A 49, 2117 (1994)] In addition to the gauge problem, there is a choice of recombination operator: velocity (or acceleration) form preferable to length form [A. Gordon, F.X. Kärtner, PRL 95, 223901 (2005), C.C. Chirilă, M.L., J. Mod. Opt. 54, 1039 (2007)] p.16
Results for harmonic generation in H + 2 a z (Ω) 2 (a.u.) 10 1 10 3 10 5 10 7 10 9 10 1 10 3 10 5 10 7 10 9 10 1 10 3 10 5 10 7 10 9 TDSE VELOCITY LENGTH ACCELERATION 0 40 80 120 160 Harmonic order θ = 40 o curves: full SFA results dashed lines: minima from TDSE recombination elements only p.17
SFA for harmonics in vibrating molecules Assume Born-Oppenheimer motion of core electrons, transition matrix element independent of internuclear distance (sufficient is d(k, R) = f(k) g(r)) Creation of a nuclear wave packet χ that evolves on the BO potential surface of the ion between t and t. p.18
SFA for harmonics in vibrating molecules Assume Born-Oppenheimer motion of core electrons, transition matrix element independent of internuclear distance (sufficient is d(k, R) = f(k) g(r)) Creation of a nuclear wave packet χ that evolves on the BO potential surface of the ion between t and t. P(t) = 2 t 0 dt E(t ) dr χ(r, 0) χ(r, t t ) d 3 p p+a(t ) x 0 0 p+a(t) exp[ is(p, t, t )] + c.c with vibrational wave packet χ(r, τ). Harmonics are sensitive to the vibrational autocorrelation function C(τ) = dr χ(r, 0) χ(r, τ) p.18
Vibrational autocorrelation function Illustration of physical mechanism: χ 0 (R) χ(r,τ) H 2 + H 2 0 1 2 3 4 5 R (a.u.) p.19
Vibrational autocorrelation function Calculate field-free evolution of a vibrational wave packet in the BO potential of H + 2 /D+ 2, i χ(r,t) t = [ 2 RM + V +BO (R) ] χ(r, t), χ(r, 0) = χ H 2 0 (R) 1.0 1.0 0.8 0.8 0.6 0.4 C(t t ) 2 0.6 0.4 0.2 0.0 0 1 2 H 2 D 2 More intense harmonics in heavier isotope D 2. 0.2 0.0 0 10 20 30 t t (fs) p.20
SFA for harmonics vibrating molecules Generalization to matrix elements depending on internuclear distance and momentum: d ion (k, R) χ H 2 0 (R) =: d ion (k) ξ k (R, t = 0) d rec (k, R) χ H 2 0 (R) =: d rec (k) η k (R, t = 0) where P x (t) = 2i t 0 dt E(t ) d 3 p C(p, t, t) d ion (p+a(t )) d rec (p+a(t)) exp[ is(p, t, t )] + c.c C(p, t, t) = dr [η p+a(t) (R, 0)] ξ p+a(t )(R, t t ) is the overlap between the evolved wave packet ξ p+a(t )(R, t t ) and the target wave packet η p+a(t) (R, 0). p.21
SFA for harmonics in vibrating molecules Approximately: ξ k (R) χ H 2 0 (R) (initial wave packet) C(p, t, t ) = dr χ H 2 0 (R) d rec(p+a(t),r) d rec (p+a(t)) χ(r, t t ) H 2 with LCAO: d rec (k, R) = N(R)k x cos(k R/2) 0 k atom Harmonics are proportional to dr χ H 2 0 (R) cos(k R/2) χ(r, t t ) 2. (This incorporates interference + vibration.) p.22
Comparison with experiment 8 fs pulses, wavelength 775 nm, intensity 2 10 14 W/cm 2 Raw data of harmonics in D 2 and H 2 p.23
Comparison with experiment 8 fs pulses, wavelength 775 nm, intensity 2 10 14 W/cm 2 Raw data of harmonics in D 2 and H 2 Ratio D 2 /H 2 Baker et al. Science 312,424 (2006) Blue: theory p.23
Conclusions It is desirable to combine nuclear motion with TDDFT Great current interest in laser-induced multielectron dynamics Next part: Learning about TDDFT and physical mechanisms from model systems p.24