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) −2.25252 −2.09369 0.15896 −   1.0 (electron) 0.6052 2 1.1653 1.7703 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) −2.890713 −2.88855 0.00223 × −3 −     4.84 10 (muon) 2.53937 2 −2.5301 0.009 1 0.1 (10 electron) −1.111452 −0.949623 0.161833

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 2b2 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