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Non-adiabatic effects in one-dimensional one- and two-electron
systems: the cases of H
2+ and H
2Alison 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, [email protected]
Outline
Motivations
Model systems: one-dimensional H2
+ and H2
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 experimentsbeyond 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)
me
mI
If << 1, the kinetic energy of
the ions is negligible: ”frozen ions”
Fictitiously vary the electron-ion
mass ratio to change the
”electron-ion coupling”
S. Takahashi and K. Takatsuka, J. Chem. Phys. 124 (2006), 1–14.
Potential Energy Surfaces (PES's)
me
m I
total≈BOA=electronicionic
Model systems: H2+ and H
2 in 1D
The exact numerical diagonalisation in real-space is feasible
Exchange symmetry of the molecular wavefunctionThe spin part is directly determined (singlet, triplet)
Soft Coulomb PotentialCoulomb potential ill-defined in 1-DR. Loudon, Am J. Phys. 27 (1959), 649-655
We use the real-space code OCTOPUS A. Castro et al., phys. stat. Sol. 243 (2006), 2465-2488http://www.tddft.org/programs/octopus/wiki/index.php/Main_page
H 2⁺ R1S 1,R2 S 2,r1 s1=−H 2⁺ R2S 2,R1S 1,r1 s1
H 2R1 S1, R2 S 2, r1 s1, r2 s2=−H 2
R2 S 2, R1 S1, r1 s1, r2 s2
H 2R1 S1, R2 S 2, r1 s1, r2 s2=−H 2
R1 S1, R2 S 2, r2 s2, r1 s1
V i nt xi−x j=qi q j
x i−x j 2a2
The 1D dihydrogen cation H2+
Hamiltonian (centre of mass frame) in atomic units (a.u.) J. R. Hiskes, Phys. Rev. 122 (1960), 1207-1217
E gs R−1R3
Non-covalent long range minimum (H+ - H)
Negligible if >>
H internal R ,=− 12I
∂2
∂R2−1
2e
∂2
∂2−1
R2 2
1− 1
R2 −2
1 1R21
I e
R=R2−R1
=r−R1R2
2
The 1D dihydrogen H2
Hamiltonian (centre of mass frame) in atomic units (a.u.)
H internal R , r ,=− 12 I
∂2
∂R2−1
2eI
∂2
∂2−1
2 e
∂2
∂r 2−1
R2 − r2
2
1− 1
R2 − r2−
2
1
− 1
R2 r2
2
1− 1
R2 r2−
2
1 1R21
1r 21
Negligible if >> I e
Non-covalent long range minimum (H – H)
E gs R−1R3
R=R2−R1 r=r2−r1
=r1r 2
2−R2R1
2
BOA validity: H2
+ case
(proton)
(muon)(10 electron)
(electron)
−3.47914
EEXACT [eV ]me
m I
5.45×10−4
4.84×10−3
1.0
EBOA[eV ] E [eV ]
EBOA
= bottom PES + zero-point energy
EEXACT
(numerical)
BOA, expansion Egs
in terms
3-D → b=1.5
1-D → b=1 (There are no contributions from rotations)
−3.74543−3.48513
−0.60522
0.00075−3.744770.0060 4
1.1653 1.7703
m I
12ℏ
me
m I
14
Bottom of the ground state PES
E=EBOA−E EXACT=a me
m Ib
b = 1.047 (1)
1-D
0.1 −2.25252 −2.09369 0.15896
BOA validity: H2 case
(proton)
(muon)(10 electron)
me
m Im I
5.45×10−4
4.84×10−3
0.1
EEXACT [eV ] EBOA[eV ] E [eV ]
−2.890713−2.539372−1.11145 2
−2.88855
−2.5301−0.949623
0.00223
0.1618330.0091
b = 0.95 (4)
No bound states for = 1 (electron)me
mI
Frozen Ion Optical Spectra H2
+
The system is perturbed by a weak ”kick” Dipole response d(t)
Continuum states (ionization)d t =−d sin eq t
1:Ground StateFirst Excited State eq
1 2 3
2:Ground StateThird Excited State3:Ground State Fifth Excited State
a bs=4 Im [ 1k∫0
t
dt e−i t f tTd t ]
(2LS)eq
∣r , t=0 ⟩=eikr∣gs ⟩
Frozen Ion Optical Spectra H2
+
For large R: H2
+ vs H
+H H+ HH+
http://www.physics.uiowa.edu/~umallik/adventure/quantumwave.html
Dynamic Ion Optical Spectra H2+
me
m I
Quicker energy transfer
(muon)
(proton)
4.84×10−3
5.45×10−4
A single peak dominates
larger asymmetry
a bs=a
2b2e−−0
2
2b2
d t =−d e−b2t 2
2 sin t
J. Mauritsson et al., PRL 105 (2010), 1–4.
Dynamic Ion Optical Spectra H2+
(proton)
(muon) 9.44110.2373
2.05191.2373
9.428610.2287
2.0211.0522
0[eV ]b [eV ]
4.84×10−3
m I ℏ [eV ]ℏb [eV ]5.45×10−4
me
m I
d t =−d e−b2 t2
2 sin t a bs=a
2b2e−−0
2
2b2
Gaussian qualitative analysis (2LS)
+
Conclusions Static case H
2+ and H
2, we find b 1 (1-D)
Dynamic case H2
+, single frequency two-level system (2LS)
dynamics for small (proton,muon)
2LS is not accurate for ;
(yet to be fully understood)
me
mI
me
mI=0.1
≈
me
mI=1
EBOA−EEXACT=a me
m Ib
Future Work
Dihydrogen H2 optical spectra from frozen and dynamic ion
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
14 September 2011
Non-adiabatic effects in one-dimensional one and two electron
systems: the cases of H
2+ and H
2Alison 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, [email protected]
THANK YOU
