Non-adiabatic contributions to the spectrum of simple molecular models: the case of the one-dimensional dihydrogen cation 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] YRM 2011, Naples, 17 May 2011 1 Outline Motivations Model system: one-dimensional (1-D) dihydrogen cation, H + 2 Results: optical spectra for frozen and dynamical ions using different ionic masses Conclusions Future work YRM 2011, Naples, 17 May 2011 2 Motivations Assess the accuracy of the Born-Oppenheimer Approximation (BOA) = total electronic ionic Fictitiously vary the electron-ion m mass ratio e mI m If e << 1, the kinetic energy of mI the ions is negligible: ”frozen ions” H + adiabatic 2 Potential Energy Surfaces (PES) YRM 2011, Naples, 17 May 2011 3 Simple model: the one dimensional dihydrogen cation 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 p R 2 e R R R 1 1 − 1 Negligible if p >> e 2 2 Van der Waals minimum Soft Coulomb Potential: Coulomb potential ill-defined in 1-D R. Loudon, Am J. Phys. 27 (1959), 649-655 The numerical diagonalisation is feasible We use the real-space code OCTOPUS 1 A. Castro et al., phys. stat. Sol 243 (2006), 2465-2488 E gs R 3 http://www.tddft.org/programs/octopus/wiki/index.php/Main_page R 4 Frozen Ion Optical Spectra The system is perturbed by a weak ”kick” Electric dipole d(t) response Time domain Frequency domain = d t d cos t Continuum states (ionization) eq 2 ”Fourier Transform d( ) ” 1 2 3 1:Ground State First Excited State eq 2:Ground State Third Excited State 5 3:Ground State Fifth Excited State Frozen Ion Optical Spectra single frequency two-level system (2LS) eq Req As m I m e m R fitted [A] [eV ] decreases the m I eq eq I R and × −4 eq eq 5.45 10 (proton) 1.1540 1 10.6297192 5 increase −3 4.84×10 (muon) 1.1559 10.6311886 8 1.0 (electron) 1.27091 11.6654661 6 BOA accuracy m e [ ] [ ] [ ] mI E EXACT eV E BOA eV E eV Bottom ground state PES mI −4 − − 5.45×10 (proton) 3.7454 3 3.7447 7 0.0007 5 − − 4.84×10 3 (muon) 3.4851 3 −3.47914 0.0060 4 1.0 (electron) −0.60522 1.1653 1.7703 1 ℏ E = bottom PES + zero-point energy BOA 2 E (numerical) EXACT 1 a = 1.7699 (1) me 4 BOA, expansion E in terms gs b = 1.066 (4) mI b = − = me E E BOA E EXACT a mI 3-D → b=1.5 1-D 1-D → b=1 (There are no 7 contributions from rotations) Electron+Ions Optical Spectra Time domain Frequency domain m e 2 2 − 2 − b t − 0 m = 2 a 2 I d t d e cos t d = e 2b 2 2 b2 − 5.45×10 4 (proton) A single peak dominates − 4.84×10 3 (muon) larger asymmetry 8 Quicker energy transfer Electron+Ions Optical Spectra ? Time domain Frequency domain − 2 b2 t 2 − 0 − 2 = a 2b d t =d e 2 cos t d e 2 2 b2 m e ℏ [ ] ℏ [ ] [ ] [ ] mI b eV eV b eV 0 eV mI −4 5.45 × 10 (proton) 1.1687 2 10.3984 5 1.228 3 10.385 3 −3 4.84 × 10 (muon) 1.890 1 9.918 3 2.003 9 9.875 9 YRM 2011, Naples, 17 May 2011 9 Conclusions Static case, we find a BOA power law exponent b ≈ 1 (1-D) Dynamic case, single frequency two-level system (2LS) m dynamics for small e (proton,muon) mI m 2LS is not good for e =1 mI Time domain Frequency domain ? 10 Future Work Same analysis for molecular hydrogen H (e-e correlation 2 included) Compare with TDDFT and Ehrenfest calculations Consider more realistic electromagnetic pulses (e.g. attosecond XUV pulse) for molecular photodissociation YRM 2011, Naples, 17 May 2011 11 Non-adiabatic contributions to the spectrum of simple molecular models: the case of the one-dimensional dihydrogen cation 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] THE END YRM 2011, Naples, 17 May 2011 12.