Non-adiabatic effects in one- dimensional one- and two-electron systems: the cases of H + and H 2 2 Alison Crawford Uranga, L. Stella, S. Kurth, and A. Rubio NanoBio Spectroscopy Group, European Theoretical Spectroscopy Facility (ETSF), Departamento de Física de Materiales, Universidad del País Vasco, San Sebastián, Spain. [email protected] Outline Motivations Model systems: one-dimensional H + and H 2 2 Results: Validity of the Born Oppenheimer Approximation Optical spectra from frozen ion calculations Optical spectra from dynamic ion calculations Conclusions Future work Motivations Interpret pump-probe attosecond experiments beyond the Born Oppenheimer Approximation G. Sansone et al., Nature Letters 465 (2010), 763–767. The Born Oppenheimer Approximation Assess the validity of the Born-Oppenheimer Approximation (BOA) ≈ = total BOA electronic ionic m If e << 1, the kinetic energy of mI the ions is negligible: ”frozen ions” Fictitiously vary the electron-ion m mass ratio e to change the mI Potential Energy Surfaces (PES's) ”electron-ion coupling” S. Takahashi and K. Takatsuka, J. Chem. Phys. 124 (2006), 1–14. Model systems: H + and H in 1D 2 2 The exact numerical diagonalisation in real-space is feasible Exchange symmetry of the molecular wavefunction The spin part is directly determined (singlet, triplet) R S R S r s =− R S R S r s H 2 ⁺ 1 1, 2 2, 1 1 H 2⁺ 2 2, 1 1, 1 1 R S R S r s r s =− R S R S r s r s H 2 1 1, 2 2, 1 1, 2 2 H 2 2 2, 1 1, 1 1, 2 2 R S R S r s r s =− R S R S r s r s H 2 1 1, 2 2, 1 1, 2 2 H 2 1 1, 2 2, 2 2, 1 1 Soft Coulomb Potential q q Coulomb potential ill-defined in 1-D V x −x = i j i nt i j 2 2 R. Loudon, Am J. Phys. 27 (1959), 649-655 − xi x j a We use the real-space code OCTOPUS A. Castro et al., phys. stat. Sol. 243 (2006), 2465-2488 http://www.tddft.org/programs/octopus/wiki/index.php/Main_page The 1D dihydrogen cation H + 2 = − R R2 R1 R R =r− 1 2 2 Hamiltonian (centre of mass frame) in atomic units (a.u.) J. R. Hiskes, Phys. Rev. 122 (1960), 1207-1217 =− 1 ∂2 − 1 ∂2 − 1 − 1 1 H internal R , 2 2 2 ∂ 2 ∂ 2 2 2 I R e R R − R 1 1 1 Negligible if I >> e 2 2 Non-covalent long range minimum (H+ - H) 1 E R− gs R3 The 1D dihydrogen H 2 = − = − R R2 R1 r r2 r1 = r1 r 2 − R2 R1 2 2 Hamiltonian (centre of mass frame) in atomic units (a.u.) =− 1 ∂2 − 1 ∂2 − 1 ∂2 − 1 − 1 H internal R , r , 2 2 2 2 ∂ 2 ∂ 2 ∂ 2 2 I R eI e r R − r R − r − 1 1 Negligible if I >> e 2 2 2 2 − 1 − 1 1 1 2 2 2 2 R r R r R 1 r 1 1 − 1 2 2 2 2 Non-covalent long range minimum (H – H) 1 E R− gs R3 BOA validity: H + case 2 m e m [ ] [ ] [ ] I E EXACT eV E BOA eV E eV mI × −4 − − Bottom of the ground state PES 5.45 10 (proton) 3.7454 3 3.7447 7 0.0007 5 −3 − − 4.84×10 (muon) 3.4851 3 3.4791 4 0.0060 4 0.1 (10 electron) −2.25252 −2.09369 0.15896 − 1.0 (electron) 0.6052 2 1.1653 1.7703 E = bottom PES + zero-point energy 1 ℏ BOA 2 E (numerical) EXACT 1 b = 1.047 (1) me 4 BOA, expansion E in terms gs mI b = − = me E E BOA E EXACT a mI 3-D → b=1.5 1-D 1-D → b=1 (There are no contributions from rotations) BOA validity: H case 2 m e [ ] [ ] [ ] mI E EXACT eV E BOA eV E eV mI − 5.45×10 4 (proton) −2.890713 −2.88855 0.00223 × −3 − 4.84 10 (muon) 2.53937 2 −2.5301 0.009 1 0.1 (10 electron) −1.111452 −0.949623 0.161833 me No bound states for = 1 (electron) mI b = 0.95 (4) Frozen Ion Optical Spectra H + 2 ∣ = 〉= ikr∣ 〉 The system is perturbed by a weak ”kick” r ,t 0 e gs Dipole response d(t) t 1 −i t t =4 Im[ ∫dt e f d t ] =− a bs k T d t d sin eq t 0 Continuum states (ionization) 1 2 3 eq (2LS) 1:Ground State First Excited State eq 2:Ground State Third Excited State 3:Ground State Fifth Excited State Frozen Ion Optical Spectra H + 2 For large R: H + vs H 2 http://www.physics.uiowa.edu/~umallik/adventure/q uantumwave.html + + H H + H H Dynamic Ion Optical Spectra H + 2 2 2 2 − me −b t − 0 =− 2 = a 2b2 m d t d e sin t a bs e I 2 b2 − 5.45×10 4 (proton) A single peak dominates − 4.84×10 3 (muon) larger asymmetry Quicker energy transfer J. Mauritsson et al., PRL 105 (2010), 1–4. Dynamic Ion Optical Spectra H + 2 Gaussian qualitative analysis (2LS) + b2 t2 − 2 − − 0 2 d t =−d e 2 sin t = a 2b a bs e 2b2 m e m ℏ [ ] ℏ [ ] b[eV ] [ ] I b eV eV 0 eV mI −4 5.45 × 10 (proton) 1.052 2 10.228 7 1.237 3 10.237 3 −3 4.84 × 10 (muon) 2.02 1 9.428 6 2.051 9 9.44 1 Conclusions b + m Static case H and H , we find b ≈ 1 (1-D) − = e 2 2 E BOA E EXACT a mI Dynamic case H +, single frequency two-level system (2LS) 2 m dynamics for small e (proton,muon) mI m m 2LS is not accurate for e = 0.1 ; e =1 mI mI (yet to be fully understood) Future Work Dihydrogen H optical spectra from frozen and dynamic ion 2 calculations Improve theoretical model for the dynamic ion calculations (asymmetry, gaussian) Perform TDDFT and Ehrenfest dynamics calculations and compare to the exact calculations Consider more realistic systems and electromagnetic pulses (pulse shapes) to interpret the experiments Non-adiabatic effects in one- dimensional one and two electron systems: the cases of H + and H 2 2 Alison Crawford Uranga, L. Stella, S. Kurth, and A. Rubio NanoBio Spectroscopy Group, European Theoretical Spectroscopy Facility (ETSF), Departamento de Física de Materiales, Universidad del País Vasco, San Sebastián, Spain. [email protected] THANK YOU 14 September 2011.