Table Of ContentStrong-field ionization of diatomic molecules and companion atoms: strong-field
approximation and tunneling theory including nuclear motion
Thomas Kim Kjeldsen and Lars Bojer Madsen
Department of Physics and Astronomy, University of Aarhus, 8000 A˚rhus C, Denmark
We present a detailed comparison of strong-field ionization of diatomic molecules and their com-
panion atoms with nearly equal ionization potentials. We perform calculations in the length and
5
velocity gauge formulations of the molecular strong-field approximation and with the molecular
0
tunneling theory, and in both cases we consider effects of nuclear motion. A comparison of our
0
results with experimental data shows that the length gauge strong-field approximation gives the
2
most reliable predictions.
n
a PACSnumbers: 33.80.Rv,32.80.Rm
J
4
1 I. INTRODUCTION 11, 12, 13] is illuminating since effects are factored out
which depend only on the binding energy. Hence a com-
]
h When a molecule is subject to an intense laser field, parison of the ratio of ionization signal of, e.g., N2 and
p a series of related strong-field processes may occur, in- itscompanionAratomallowsonetostudyfairlydirectly
- effectsofmolecularsymmetryandro-vibrationalmotion.
m cluding ionization, dissociation and high-harmonic gen-
In Ref. [3] the status of theory versus experiments was
eration. As for atoms, it is essential to obtain a detailed
o discussed: The MO-ADK theory [3] gave results in sat-
understanding of the initial, single-ionization process in
t isfactoryagreementwith experiments for molecules with
a ordertodescribethesubsequentevolutionofthesystem.
s. Fully ab initio calculations of the strong-field ionization suppressed ionization: D2:Ar [10, 13], O2:Xe [8, 11] and
c of any molecule more complicated than H are impossi- without: N2:Ar [11].
i 2
s ble in the foreseeable future. Hence, theories of general FortheF2:Arratio,theMO-ADKtheorypredictssup-
y practical use have to rely on theoretical modelling and pression of the molecular signal while the experimental
h it is the purpose of the present work to investigate the data [12] does not. For the F2 molecule, however, the
p
accuracy of such models. While the quality of ab ini- totalenergy obtainedfrom a Hartree-Fock(HF) calcula-
[
tiocalculationsmaybecheckednumericallybystudiesof tionofthe molecularsystemis higherthanthe energyof
2 convergenceandbyidentityofresultsindifferentgauges, the separated atoms [14]. This means that electron cor-
v the quality of an approximate model has to be checked relation effects are important for the proper description
1
against a more accurate model or experimental data. of the binding of F2, and, accordingly, the one-electron
2
modeltakenas the startingpointforthe MO-ADK(and
2 In the study of strong-field ionization of molecules,
the MO-SFA) theory, is not applicable. The models dis-
1 three models are widely used. These are the velocity
1 gauge (VG) molecular strong-field approximation (MO- cussed above, and in the present paper, can only be
4 SFA VG) [1], the length gauge (LG) MO-SFA (MO-SFA meaningfully applied on molecular systems where effects
0 of electron-electron correlation are small [15]. Methods
LG) [2], and the molecular tunneling theory [3]. The
/ which do take electron-electron correlation into account
s latter theory is an extension of the atomic Ammosov-
c arepresentlyunderdevelopment[16,17]butthesemeth-
Delone-Krainov (ADK) tunneling theory [4] generalized
si to take into account the non-spherical symmetry of the ods are not yet applicable to molecules interacting with
y oscillating fields in geometries without cylindrical sym-
molecular system and it is referred to as the MO-ADK
h metry.
theory. The MO-SFA VG and MO-SFA LG theories
p
are the velocity gauge and length gauge versions of the The MO-SFA VG was compared with experiments in
:
v Keldysh-Faisal-Reiss (KFR) atomic theories [5, 6, 7] ap- Ref. [1]. The theory was shown to predict suppressed
i
X propriately modified to the molecular case. In short, the ionization of O2:Xe [8, 11] and to predict ‘absence’ of
r MO-SFA theories are based on an S-matrix formulation suppressed ionization in N2:Ar [11]. In Ref. [18], the
a where one considers the transition from a field-free ini- predictions of the MO-SFA VG were compared with ex-
tial state to a Volkov final state, i.e., the state of a free perimental above-threshold-ionization spectra of O2 (πg
electron in the laser field. symmetry)and N2 (σg symmetry)and for these systems
To assess the quality of the models, predictions have the theory gave the correct qualitative predictions.
beencomparedwithexperimentaldata(see,e.g.,Ref.[3] In general, the MO-SFA VG approximation predicts
and references therein). In particular, data has been suppressedionizationfordiatomicmolecules withanan-
studied in detail which compares ionization yields of di- tibonding highest occupied molecular orbital (HOMO)
atomic molecules with yields of atoms having nearly the and it predicts no suppression of the ionization of di-
same electronic binding as the molecules – the so-called atomswithabondingHOMO.Thesegeneralpredictions
companion atoms. The ratio of ion signal data of di- arebasedonananalysisoftheMO-SFAVGrateinterms
atomic molecules and their companion atoms [8, 9, 10, of linear combinations of atomic orbitals where the an-
2
tibonding [bonding]rate is proportionalto sin2(q R/2) A. Molecular strong-field approximation including
cos2(q R/2) andq R 1whereq isthemom·entum nuclear motion in the Born-Oppenheimer
of the ou·tgoing electro·n a≪nd R the internuclear coordi- approximation
(cid:2) (cid:3)
nate. For example,the MO-SFAVG approximationpre-
dicts suppressed ionization for O2 with a HOMO of πg IntheCoulombgaugeandinthedipoleapproximation
symmetry and no suppression of the D2:Ar signal. This the linearly polarizedlaserfield maybe describedby the
latter prediction is in contrast with experimental obser- vectorpotentialA(t)=A cosωt,whereω istheangular
0
vation[13] and MO-ADK theory [3], andwe shallreturn frequency. Thecorrespondingelectricfieldisobtainedas
to this discrepancy below. F(t) = ∂ A(t), i.e., F(t) = F sinωt. The interaction
t 0
−
between the field and an N-electron system is [atomic
In view of the above discussion at least two questions units (~=m =e=a =1) are used throughout]
e 0
need to be investigated in more detail. Firstly, what is
the quantitative prediction of the MO-SFA in the D2:Ar N A2(t)
case? Secondly, is it possible to carry out a calcula- V(VG)(t)= A(t) p + , (1)
j
· 2
tioninamoreaccurateapproximationwhichshedssome j=1
X
light on the discrepancy between the MO-ADK calcula-
tionsofRef.[3]andtheexperimentalresultsofRef.[13]? in the velocity gauge and
To address related questions we have recently developed
N
the necessary tools for computations in the velocity and
V(LG)(t)= r F(t), (2)
length gauge formulations of the MO-SFA, and in addi- j ·
j=1
tion we have set up a programme for the evaluation of X
the molecular tunneling theory [2, 15]. Our LG and VG in the length gauge. In either gauge, we express the an-
versionsoftheMO-SFAaregeneralizationsoftheatomic gular differential dW/dqˆand total W rates as sums over
adiabatic theory [19]. Our formulation of the MO-ADK n-photon absorptions [19]
theory follows Ref. [3], but our method of extracting the
angular coefficients of relevance for the evaluation of the dW ∞
rateisdifferent[2,15]. Inthiswork,weextendourprevi- dqˆ =2π |Aqn |2 qn, (3)
ousanalysis,bytakingeffectsofnuclearmotionexplicitly nX=n0
into account and we show that the inclusion of nuclear
vibrationsmayleadto a significantlylowerrate. Forthe
∞
molecules considered in this paper nuclear motion is of W =2π A 2 q dqˆ, (4)
qn n
significant importance for D and NO while it leads to | |
small effects in the final resu2lts for the other molecules nX=n0Z
studied. wherethe transitionamplitudescorrespondingto the in-
teraction V(c)(t), (c= VG,LG ),
The paper is organized as follows. In Sec. II, we de- { }
scribethetheoriesandprovideaqualitativediscussionof 1 T
A(c) = Ψ V(c)(t) Ψ dt, (5)
the effects of nuclear motion. In Sec. III, we give some qn T h f | | ii
calculational details, in Sec. IV we present the results of Z0
our calculations, and in Sec. V, we conclude. involves integration over one period of the field T = 2π,
ω
and designates integration over both the electronic
andnhu||cilearcoordinates. HereΨ describesthemolecular
i
initialstate andΨ is the final state describingthe state
f
ofthe residualionandthe free electroninthe laserfield.
In Eqs. (3)-(4), n is the minimum number of photons
0
neededtoreachthecontinuum,andthemomentumq is
n
II. THEORY given by energy conservation. In the Born-Oppenheimer
(BO)approximationq isdeterminedbyEq.(10)below.
n
In the SFA we approximatethe initial state by a field-
InthissectionwedescribetheMO-SFAandMO-ADK freemolecularstate. Inthe BO approximationthis state
theories and discuss how to account for effects of nu- is a product of an electronic state and a ro-vibrational
clear motion. In the MO-SFA one considers a state to state labelled by ν , J . The electronic and vibrational
i i
statetransitionwherethemolecularstatesaregenerated states are typically the respective ground states. The
from moving nuclei. Contrary, in the MO-ADK there is rotational periods of the diatomic molecules are much
no detailed specification of the final state in the strong- longer than typical experimental pulse durations and
field ionization process, and in order to maintain a qua- therefore the rotational degrees of freedom may be ne-
sistatic tunneling picture one must fix the nuclei at an glected. The total energy of the initial state is
internuclear distance R and let the electron move in the
R-dependent potential. E =Ee(R )+E , (6)
i i 0 νi
3
where Ee(R ) is the electronic eigenenergy at the inter- HOMO is affected. The final state is then
i 0
nuclear equilibrium distance R and E is the vibra-
0 νi
tional eigenenergy of the nuclear Hamiltonian. If we ap- 1
Ψ = det ψ (r )ψ (r )...ψ (r ,t) (8)
proximatethe electronicpartofthe initialwavefunction f √N! | 1 1 2 2 V N |R0
by the single-determinant HF wave function, the corre- χ+(R)e−iEf+t,
sponding initial molecular wave function is × νf
1 whereψ isa(2π)−3/2 normalizedVolkovwavefunction
Ψ = det ψ (r )ψ (r )...ψ (r ) (7) V
i √N! | 1 1 2 2 N N |R0 and where the superscripts “+” denote the ionic state.
χ (R)e−iEit, Thetime-averagedenergyoftheelectroninthelaserfield
× νi is q2/2+U , and the total final-state energy is
n p
where χ (R) is the initialvibrationalwavefunction and
νi
the ψ ’s are orthogonal single-electron wave functions. q2
j E =Ee,+(R )+E+ + n +U , (9)
The electronic wave function is evaluated at the nuclear f f 0 νf 2 p
equilibrium distance R since we assume, consistently
0
with the BO picture, that it will be a slowly varying where U =F2/4ω2 is the ponderomotive potential, and
p 0
function of the internuclear distance. We have checked where the finalstate momentum q is determinedby en-
n
that the results are insensitive to this approximation. ergy conservation nω =E E , i.e.,
f i
We seek the transition amplitude to a single-electron −
Volkov state and a definite vibrational and electronic
eigenstate of the molecular ion. The application of a qn = 2(nω−(Efe,+(R0)+Eν+f −Eie(R0)−Eνi)−Up).
Volkov wave in the final state means that the electron- q (10)
ioninteractionisneglected. Additionally,weassumethat The transition amplitude of Eq. (5) can now be written
the electronic state of the ion is unrelaxed, i.e., only the as (c= VG,LG )
{ }
1 T
A(c) = χ+(R)ψ (r ,t) V(c)(t) ψ (r ;R )χ (R) exp i Ee,+(R )+E+ Ee(R ) E t dt, (11)
qn T h νf V N | N | N N 0 νi i f 0 νf − i 0 − νi
Z0 h (cid:16) (cid:17) i
where the N-electronmatrix element of the one-electron where S is the Franck-Condon(FC) factor
νf,νi
operators, Eqs. (1) and (2), was been simplified by the
Slater-Condon rules [20] and where V(c)(t) is the tran-
N
sition operator for electron N. The integration over nu-
∗
clear coordinates can be performed immediately to give Sνf,νi = χ+νf(R) χνi(R)dR. (13)
Z h i
1 T
A(c) = S ψ∗(r ,t)V(c)(t)ψ (r ;R )dr
qn νf,νiT V N N N N 0 N
Z0 Z From Eq. (12), we find the following explicit expressions
ei Efe,+(R0)+EνNf,+−Eie(R0)−EνNi tdt, (12) for the amplitudes
× (cid:16) (cid:17)
1 T [q +A(t)]2 U
A(LG) =S E n ψ˜ (q +A(t))expi nωt+q α sin(ωt)+ p sin(2ωt) t (14)
qn νf,νiT Z0 − b− 2 ! N n (cid:18) n· 0 2ω (cid:19)
q2 1 T U
A(VG) =S E n ψ˜ (q ) expi nωt+q α sin(ωt)+ p sin(2ωt) t, (15)
qn νf,νi − b− 2 N n T n· 0 2ω
(cid:18) (cid:19) Z0 (cid:18) (cid:19)
In Eqs. (14)-(15), α = A /ω is the quiver radius, space wave function of the HOMO, and E is the energy
0 0 b
ψ˜ (q) = (2π)−3/2 dre−iq·rψ (r) is the momentum
N N
R
4
difference between the final and initial state identicalthenthevibrationaleigenstateswillbeidentical
too. Theorthogonalityofthenuclearwavefunctionsthen
Eb =Efe,+(R0)+Eν+f −Eie(R0)−Eνi. (16) assuresthatonlyasingleFCwillbenonzero. Wemayes-
timatetheimportanceofnuclearvibrationsusingmolec-
When we compare Eqs. (14) and (15), we see that the
ularorbitaltheoryandbyconsideringthetypeofvalence
length gauge formulation accounts for a superimposed
orbitals. If the valence orbital is nearly non-bonding as,
quiver motion in momentum space of the bound state
e.g., in N , the bonding properties of the molecular ion
2
electron via the presence of the A(t) term in the argu-
will be approximatelyequal to the bonding propertiesof
ment of the momentum space wave function. Such an
neutralmolecule andtransitionsbetweenthe vibrational
effect is not present in the velocity gauge amplitude.
ground states of the molecule and the ion dominate. In
To account for the Coulomb interaction between the
the case of a bonding HOMO, e.g., as in D , the bond-
2
outgoingelectronandthe residualmolecularion one has
ing of the ion will be weakened and transitions to many
typically introduced a Coulomb correction factor. In
vibrational states will occur.
the velocity gauge,this factor is C(VG) = κ3/F 2Zion/κ
Coul 0
with κ = √2E [1] and Z the charge of the residual
b ion (cid:0) (cid:1)
ion, while in the length gauge C(LG) = 2κ3/F 2Zion/κ B. Molecular ADK theory
Coul 0
[21]. Both correctionfactors were derived for the case of
strong-field ionization of atoms, and hen(cid:0)ce do n(cid:1)ot take The tunneling theory [3, 4] relies on the assumption
into account the molecular symmetry. In our evaluation thatatanygiveninstantoftime thesystemwillrespond
of rates, we have found that much more precise results totheexternallaserfieldasifitwereastaticelectricfield.
are obtained in the length gauge with C(LG) =1 [2, 15]. Therateofionizationintheoscillatingfieldmaythenbe
Coul determined by time-averaged static rates. Whether this
We explain the absense of a Coulomb interaction in the
approach is reasonable or not depends on the value of
lengthgaugebythefactthatthetransitiontothecontin-
the Keldysh parameter γ = √2E ω [5] with γ 1 in
uum occurs at large distances. In this spatial region the bF0 ≪
thetunnelingregime. Wewillonlyshowresultsfromthe
laser-electroninteractionisstrongerthantheelectron-ion
MO-ADK theory in the intensity regions corresponding
interactionandtheVolkovstateisagoodapproximation
to γ 1.
forthefinalstate. InadditiontotheCoulombcorrection ≤
The tunneling rate of diatomic molecules can be de-
factor, we multiply the rates by the number of equiva-
termined once the field-free asymptotic wave function is
lent electrons in the HOMO. Finally, to obtain the total
known. In a body-fixed frame, labeled by superscript B
ionization rate of the molecules, we must sum the con-
and a z axis directed along the internuclear axis, this
tributions from each vibrational level in the final state.
function has the asymptotic Coulomb form
In the velocity gauge our result is equivalent to that of
Ref. [22]. For the noble gas atoms with filled p shells we
ψB(r) rZion/κ−1e−κr C Y (rˆ), (17)
sumtheratesfromeachmagneticsub-statetoobtainthe N ∼ lm lm
l
total rate of ionization. X
where m is the projection of the angular momentum on
the internuclearaxis,and whereC areexpansioncoef-
lm
1. Qualitative discussion of the effect of nuclear motion ficients.
FromtheasymptoticformofEq.(17),thetotalioniza-
The transition amplitude of Eq. (12) consists of two tionrateinastatic(DC)fieldinthepositivelaboratory-
factors, namely the Franck-Condon (FC) factor and an fixed Z direction is calculated as in the atomic case
electronicmatrixelementandbothfactorsdependonthe [21, 23, 24], and the result is [3]
vibrational levels considered. The rates to each vibra-
B(m′)2
tional levelare therefore not just proportionalto the FC W (F ) = | | (18)
factors. Insteadtherelativepopulationsinthelowerfinal stat 0 m′ 2|m′||m′|!κ2Zion/κ−1
X
vibrational states are enhanced compared with the dis- 2κ3 2Zion/κ−|m′|−1 2κ3
tributionobtainedfromtheFCfactorsalone[22]because exp ,
the electronicmatrix elementis favouredby the smallest × (cid:18) F0 (cid:19) (cid:18)−3F0(cid:19)
energy differences. When including vibrations, the total
where
ratesummedoverallfinalvibrationalstateswilltherefore
typically be smaller than if the vibrational ground state B(m′) = ( 1)(|m′|+m′)/2 (2l+1)(l+|m′|)!
oftheionhadbeengiventheweightofunity. Thislatter − s 2(l m′ )!
methodisnearlyequivalenttofixingthe nuclei[compare Xl −| |
Eqs. (14)-(15) with Eqs. (5)-(6) of Ref. [2]]. C d(l) (θ). (19)
× lm m′m
The importance of the inclusion of nuclear vibrations
will depend on the properties of the neutral molecule Here d(l) (θ) is the middle term of Wigner’s rotation
m′m
and the molecular ion. If their potential curves are only function[25]andθistheanglebetweenthefielddirection
shifted with respect to each other but otherwise exactly and the internuclear axis.
5
In a slowly varying field, the ionization rate is found
Table I: The molecular and atomic properties necessary for
by averaging the DC rate over an optical cycle [15]
theevaluationofthepresentMO-SFALGandtheMO-ADK
W 3F0Ws+tat(F0)+Ws−tat(F0), (20) ttihaeloaryn.d RIp0 iiss tthheeeeqxupileirbirmiuemntadlisatadniacbea[t2i6c].ioFnuizrathtieornmoproetewne-
≈rπκ3 2 givetheangularcoefficientsClmfromourHartree-Fockbased
calculation together with the coefficients from Ref. [3]. We
where W± (F ) are the DC rates for the positive and have chosen the origin at the geometrical midpoint. Franck-
stat 0
negative field directions with respect to the Z direc- Condon factors and vibrational energies can be found in the
tion. When the field points in the negative Z direction, references indicated after each molecular species.
Eqs. (18) and (19) must be modified by the substitution
C ( 1)lC [15]. We see from Eqs. (18)-(20) that Ip R0 C0m C1m C2m C3m C4m
lm → − lm (eV) (˚A)
oneonlyneedstoknowtheC coefficientsinordertobe
lm
able to evaluate the tunneling formula analytically. See D2 (σg) [32] 15.47 0.742 2.44 0.14 0.00
alsoRef. [15]for a generalizationofthe tunneling theory 2.51 0.06 0.00 [3]
to molecules with more than two nuclei. N2 (σg) [33] 15.58 1.098 3.46 1.64 0.12
Thetunnelingtheorycanbeextendedtoincludeeffects 2.02 0.78 0.04 [3]
ofvibrations. Inthequasistaticpictureallpotentialsseen O2 (πg) [34] 12.03 1.208 1.04 0.07
bytheactiveelectronshouldberegardedasbeingstatic. 0.62 0.03 [3]
Thismeansthatthenucleiarefixedandtheheightofthe S2 (πg) [35]a 9.36 1.889 1.46 0.24
tunnelingbarrierwilldependontheinternucleardistance
0.81 0.07 [3]
chosen. We then calculate the rate for each value of R
CO (σ) [34] 14.01 1.128 -3.93 2.79 -1.59 0.31 -0.09
andweighttheseR-dependentratesbytheRprobability
1.43 0.76 0.28 0.02 [3]
distributionobtainedfromthe nuclearwavefunction [3].
As an approximate binding energy we take the energy NO (π) [34] 9.26 1.151 -0.25 0.82 -0.06 0.04
difference between the potential curves of the ion and 0.22 0.41 0.01 [3]
neutral molecule as calculated by HF theory. The vibra- SO (π) [36]a 10.29 1.481 1.09 -1.25 0.34 -0.12
tionalwavefunctions are constructedfromthe harmonic 0.41 -0.31 0.01 [3]
approximation to experimental potential curves [26]. As
Ar (p) 15.76 2.51
demonstrated in Sec. IV the effect of this R-dependent
2.44 [3]
weighting is quite small. Note in passing that recently
Kr (p) 14.00 2.59
vibrational distributions were measured and calculated
2.49 [3]
with tunneling theory [27].
Xe (p) 12.13 2.72
2.57 [3]
III. CALCULATIONAL DETAILS
aBasedonphotoelectron spectrum
In order to determine the angular coefficients C we
lm
evaluate the ground state wave function in the HF ap-
If we use the experimental ionization potential in the
proximation. We seek an accurate description of the
calculation of κ, the HOMO is not guaranteed to follow
orbitals at large distances with the correct exponen-
the correct asymptotic form of Eq. (17). The asymp-
tial behaviour. For this purpose we find in general the
usual expansion in an atomic Gaussian basis to be inad- totic form will be similar but with κHF = 2εHOMO
| |
substituted for κ with ε the eigenvalue of the one-
equate. Instead we solve the HF equations fully numer- HOMO
p
ically for the diatomic molecules [28] and for the atoms electron HF (Roothaan) equation. For the highly corre-
[29, 30]. After having obtained the groundstate orbitals lated F− ion, for example, the two different values of κ
we project the highest occupied orbital on the spherical differ by 20%and the applicationof a wrong asymptotic
harmonics and match the resulting radial functions to wavefunction will introduce a largeerrorwhen calculat-
the form ClmrZion/κ−1e−κr treating the angular coeffi- ing the MO-SFA LG ionization rates [31]. For the sys-
cients as fitting parameters. We give the C ’s obtained tems considered here the largest difference between the
lm
by this procedure in Table I. Since orbitals from the experimentalκ andκHF is 10%for NO andless than5%
HF calculation are optimal within the independent par- for the remainingsystems. Thus we conclude thatcorre-
ticle approximation, the HF C coefficients should be lationeffectsdonotaffecttheouterelectronsignificantly
lm
more accurate than the multiple scattering coefficients for the systems considered here. We believe that the
reported in Ref. [3]. smallness of the differences between κ and κHF justifies
the application of the single-active electron models.
The knowledge of the coefficents C is sufficient to
lm
evaluate the MO-SFA LG rate accurately [15]. For the With the ionization rates at hand, we calculate the
evaluationofthe MO-SFAVGtransitionamplitudes, we ion signals by integrating over the temporal and spatial
make a numeric Fourier transform of the HOMO. profile of the pulse [15, 37]. In all experiments discussed
6
1012
(a) MO-SFA LG
4.0 (a) (b)
+Ar MO-MSOFA-A VDGK 1011
+gnal ratio N / 2 123...000 Guo et al. (1998) -1n rate (s) 1110001890
Si zatio 107
ni
(b) DeWitt et al. (2001) o
+ 4.0 I 106
Ar
+ / 3.0 105
2
N
gnal ratio 12..00 1041013 Intensity (Wcm-21)0 1 4 1013Intensity (Wcm - 2 ) 1014
Si
0.0 Figure2: TheionizationratesofN2andAr(dotted)atinten-
1014 1015 sitiesaroundthefirstchannelclosingsobtainedfrom(a)MO-
Intensity (Wcm-2) SFA LG and (b) MO-SFA VG. For N2, we give the rates at
parallel(solid)andatperpendiculargeometry(long-dashed).
The wavelength is 800nm and the intensities at which the
Figure1: IntensitydependentratiosbetweentheyieldsofN2 channels close for N2 are indicated by the arrows. Ar has
and Ar ions. In both panels the laser wavelength is 800nm. a slightly higher ionization potential and the corresponding
In panel(a),thepulseduration is 30fs and theexperimental intensities lie at bit lower.
data is from Ref. [11]. In panel (b), the pulse duration is
100fs and the experimental data is from Ref. [12]. The thin
linesarecalculations with fixednucleiandthethicklinesare cal. These findings are in reasonableagreementwith the
thecorresponding calculations includingvibrations. experimental data from Refs. [11, 12]. The differences
between our MO-ADK results and the results of Ref. [3]
lie in the values of the angular C coefficents. We find
lm
here,theionswerecollectedthroughasmallpinholenear the ratio to be slightly lower than unity in agreement
the beam waist and the spatial integration should be re- with Ref. [11] [Fig. 1 (a)]. The velocity gauge MO-SFA
strictedtothisregion. Asthesignalssaturate,thesignal predicts the ion yield of N to be somewhat larger than
2
ratioswill approachunity for sucha setup. We calculate for Ar in accordance with Ref. [1]. We find an oscillat-
the rates and signals for each molecular orientation and ing behaviour of the signal ratio in the velocity gauge
averagethe signals overorientationsin orderto simulate which is not supported by the experimental data. The
randomly oriented ensembles. origin of this artifact lies in the concept of channel clos-
ing: At low intensities continuum-continuum transitions
are very weak. The rate of ionization is then typically
dominated by the rate which originates from the low-
IV. RESULTS AND DISCUSSION
est number of photon absorptions, n . As the intensity
0
increases the ponderomotive potential rises and eventu-
A. N2:Ar ally leads to the closing of the n channel. The effect of
0
channel closing on the total ionization rate depends on
Several experiments have been performed on N2 and the relative importance of the n0 processcomparedwith
its companion atom Ar [9, 11, 12, 18], and there are the excess-photon processes. As the intensity increases
significant disagreements between the results. The yield and a channel closing approaches, the momentum corre-
ratios were estimated to be N2:Ar ≈ 0.7 [11], 0.2 [9], 1 sponding to n0 photon absorptionswill become small. If
[18]and1.7[12],respectively. Previoustheoreticalcalcu- the HOMO does not contain an l = 0 component, then
lationswiththeMO-SFAVGmodelpredictedtheN2:Ar ψ˜N 0 as q 0, and for such systems the MO-SFA
→ →
ratio to be above unity [1] while MO-ADK calculations VG amplitude, Eq. (15), will be suppressed. Near the
predicted the ratio to be 0.4 0.6. In Ref. [3] it was channel closings the contribution from the n channel is
− 0
shownthatthe experimentaldifferencescouldnotbe ex- thus small and the total rate will not be significantly af-
plainedbydifferencesinpulselengthsandthepossibility fectedwhenthis channelcloses. If onthe otherhandthe
of dynamical alignment. HOMO contains an l = 0 component, the low momenta
In Fig.1 we show our calculationsfor the N :Ar ratio. willgenerallybefavouredandthen channelgivesalarge
2 0
First,wenotethattheresultsfromeachmodelarenearly contributionof the totalrate. In this case,the closingof
independent of pulse duration. In the length gauge we achannelwillthereforeleadtoanabruptdecreaseofthe
predict the ion yields of N and Ar to be nearly identi- total rate. These differences were previously mentioned
2
7
++ / Ar2 3.0 (a) Wells MMetOO a-M-SlS. OFF(2A-AA0 V0DL2GGK) ++O / Xe2 111...024 (a) Guo MMetOO a-M-SlS. OFF(1A-AA9 V9DL8GGK)
al ratio D 12..00 nal ratio 000...468
n g 0.2
Sig Si 0.0
+e 1.4 (b) DeWitt et al. (2001)
++ / Ar 3.0 (b) Talebpour et al. (1998) +o O / X2 011...802
D2 2.0 ati 0.6
atio nal r 0.4
al r 1.0 Sig 0.2
n 0.0
Sig +e 1.4 (c) Talebpour et al. (1996)
X 1.2
0.0 1014 1015 +O / 2 1.0
Intensity (Wcm-2) o 0.8
ati 0.6
al r 0.4
n
Figure3: IntensitydependentratiosbetweentheyieldsofD2 Sig 0.2
and Ar ions. In both panels the laser wavelength is 800nm. 0.0
1014 1015
Thethinlinesarecalculationswithfixednucleiandthethick
lines are the corresponding calculations including vibrations. Intensity (Wcm-2)
Inpanel(a),thepulsedurationis100fsandtheexperimental
dataisfromRef.[13]. Inpanel(b),thepulsedurationis200fs
and theexperimental datais from Ref. [10]. Figure4: IntensitydependentratiosbetweentheyieldsofO2
and Xe ions. In all panels the laser wavelength is 800nm.
Thethinlinesarecalculationswithfixednucleiandthethick
lines are the corresponding calculations including vibrations.
in Ref. [38]. In the length gauge, the n channel will be
0 Inpanel (a),thepulseduration is 30fs andtheexperimental
importantregardlessofthetypeoforbitalduetothepre-
data is from Ref. [11]. In panel (b), the pulse duration is
cense of A(t) in the argument of the momentum space
100fs and the experimental data is from Ref. [12]. In panel
wavefunction[seeEq.(15)]andwealwaysfindabruptde-
(c) the pulse duration is 220fs and the experimental data is
creasesofthe rateacrossachannelclosing. We showthe from Ref. [8].
various effects described above in Fig. 2. The HOMO of
N isaσ (l =0component)orbitalandcorrespondingly
2 g
we find a local minimum in the rate around the channel
closings. The highest occupied atomic orbitals of Ar are
the degenerate p (l =1) orbitals, and we find local min-
imainthelengthgaugeratewhilethevelocitygaugerate
correctly predict suppression. At high intensities the ra-
increasessmoothly nearthe channelclosings. The differ-
tio will approach unity as both signals saturate. The
ences between the parallel and perpendicalar geometries
MO-SFALG model has some problems in predicting the
of N in the two gauges was explained in Ref. [2].
2 correct intensity at which this saturation occurs, but we
notethattheexperimentsdonotagreeonthesaturation
intensityeither. Thelongerpulselengthshouldbeequiv-
B. D2:Ar
alent to a lower saturation intensity but this is clearly
not the case when comparing the experimental data in
Two experiments have been reported on D and Ar
2 Figs. 3 (a) and (b).
[10, 13] and both showed a suppression of the D sig-
2
nal compared with the Ar signal. As mentioned in the
introduction the MO-ADK calculations reproduced this We see that the inclusion of nuclear vibrations reduce
result [3] while the MO-SFA VG model predicts absence theMO-SFAratiosbyasignificantfactorcomparedwith
ofsuppressionbecause the HOMO ofD is a bonding σ thefixednucleicalculations. Theoriginofthiseffectwas
2 g
orbital. discussed in Sec. IIA1. The point is simply that the
InFig.3,weshowourcalculationsandtheexperimen- inclusionofvibrationalmotionwillreducethe ratewhen
talsetsofdata. Wedoindeedfindabsenceofsuppressed themoleculeandthemolecularionhavedifferentbonding
ionization when we use the MO-SFA VG model. On the properties and transitions to many different vibrational
other hand the MO-SFA LG andMO-ADK models both states occur.
8
C. O2:Xe 4.0
++ / Kr 3.0 (a) Wells MMetOO a--SlS. FF(2AA0 V0L2GG)
Suppressed ionization of O :Xe has been observed re- O
2 C
peatedly [8, 11, 12]. Theoretically, this was explained o 2.0
in the MO-SFA VG by the antibonding character of the ati
πg HOMO of O2 [1]. Another explanation within the nal r 1.0
g
MO-ADK model was given in Ref. [3] where the inter- Si 0.0
pretation was based on the asymptotic charge density of 1013 1014
the πg HOMO. At some molecular orientations the elec- Intensity (Wcm-2)
tronic density will be preferentially perpendicular to the
polarization axis and the rate of ionization will then be +Kr (b) MO-ADK
very small. Finally in Ref. [39] it was proposed that nu- + / 4.0
O
clear dynamics could be responsible for the suppression. C 3.0
In Fig. 4, we show the experimental data together with atio 2.0
otiuornscaslhcouwlatsiuopnps.resCseledariolyn,izaalltioexnpoefriOm2e.ntBs eafnodrecsaaltcuurlaa-- gnal r 1.0
tion effects become important the theoretical MO-SFA Si 0.0
VG predicts strongest suppression with the ratio below 5x1013 1014 2x1014
0.01 while the ratio in MO-SFA LG is 0.03. At low Intensity (Wcm-2)
≈
intensities,theexperimentalratiosarescatteredbetween
0.02 0.20whichagainmakesaquantitativecomparison
− Figure 5: Intensity dependent ratios between the yields of
with theories difficult. In the tunneling regime, beyond
CO and Kr ions. In panel (a) thelaser wavelength is 800nm
the intensity of 1014Wcm−2, the MO-ADK model pre-
and the pulse duration is 100fs. In panel (b) the laser wave-
dicts that saturation effects are already important and length is 1365nm and the pulse duration is 80fs. In both
therebythedegreeofsuppressionismasked. Experimen- panelstheexperimentaldataisfromRef.[13]. Thethinlines
tally,saturationsetsinatmuchhigherintensitiesaround are calculations with fixed nuclei and the thick lines are the
2 1014Wcm−2, in good agreement with the MO-SFA corresponding calculations including vibrations.
×
theories.
E. Molecules without companion atoms
D. CO:Kr In Ref. [13] the ion signal ratios of the pairs S2:Xe,
NO:Xe, and SO:Xe were measured. Common to these
pairs is that the ionization potential of Xe is somewhat
TheionsignalratioofCO:KrwasmeasuredinRef.[13] higher than the ionization potentials of the molecules.
and found to be around one half. In Figs. 5 (a) and The ratios measured are thus the results of both the
(b) we show the results at 800nm and 1365nm, respec- structural differences (orientation, electronic wave func-
tively. Under the experminental conditions at 800nm, tions) and the difference between the binding ener-
the Keldysh parameter exceeds unity. Thus we cannot gies. Theexperimentswereperformedatthe wavelength
rely on the MO-ADK theory and in Fig. 5 (a) we show 800nm with γ > 1, Fig. 6, and at 1365nm with γ < 1,
only MO-SFAcalculations. Boththeoriespredictthe ra- Fig. 7.
tio to be slightly larger than one. In the MO-SFA VG
The electronic structure of S is similar to O and
2 2
model, we findthatthe ratiodepends onthe intensityin
we should therefore expect the ionization of S to be
2
asimilarwayasthe N :Arratio,andsincethe electronic
2 suppressed too. Our MO-SFA calculations, shown in
structures of CO and Kr are nearly identical to N and
2 Fig. 6 (a), predict the S :Xe ratio to be much higher
2
Ar,respectively,wecanexplaintheoscillatingbehaviour
than measured. Experimentally, the ratio was deter-
by the same channel-closing argument (see Sec. IVA).
mined to be around unity. This indicates that ioniza-
At 1365nm and in the intensity range of Fig. 5 (b), tion of S would indeed be suppressed if compared with
2
we would expect to be in the tunneling regime. De- ahypotheticalcompanionion. InFigs.6(b)-(c)weshow
spite the fact that the tunneling theory should be ap- the calculationsfor SO:Xe andNO:Xe andfind a similar
plicable, the MO-ADK model predicts the ratio to be disagreement between theory and experiment for these
somewhat too large. Finally we note that the result of pairs. The two previous attempts [3, 13] to explain the
our MO-ADK calculation is approximately an order of ratiosinthisintensityregimeoverestimatedtheratiosby
magnitude largerthan the resultfrom Ref. [3]. The only 3 5 orders of magnitude. Both calculations were based
−
differences between these two calculations are the coef- on tunneling models – in Ref. [13] a purely atomic ADK
ficients C and the time-averaging of the static field model was used and in Ref. [3] the MO-ADK model was
l,m=0
rate, our Eq. (20) as compared with Eq. (10) of Ref. [3]. applied. OurpresentMO-SFAcalculationsincludingnu-
9
3.5
+e 103 (a) MO-SFA LG +e 3 (a) MO-SFA LG
X MO-SFA VG X MO-SFA VG
+S / 2 102 Wells et al. (2002) +S / 2 2 .25 Wells et aMl. O(2-A00D2K)
atio 101 atio 1.5
nal r 100 nal r 1
g g 0.5
Si Si
10-1 0
+Xe 103 (b) +Xe 2 (b)
+O / 102 +O / 1.5
S S
atio 101 atio 1
nal r 100 nal r 0.5
g g
Si 10-1 Si 0
+Xe 103 (c) +Xe 2.5 (c)
+O / 102 +O / 2
N N 1.5
atio 101 atio 1
nal r 100 nal r 0.5
g g
Si 10-1 Si 0
1013 1014 1 2 3 4 5 x1014
Intensity (Wcm-2) Intensity (Wcm-2)
Figure 6: Intensity dependent ratios between the ion yields Figure 7: Intensity dependent ratios between the ion yields
of (a) S2:Xe, (b) SO:Xe and (c) NO:Xe. In all panels the of (a) S2:Xe, (b) SO:Xe and (c) NO:Xe. In all panels the
laser wavelength is 800nm, the pulse duration is 100fs and laser wavelength is 1365nm, the pulse duration is 80fs and
the experimental data is from Ref. [13]. The thin lines are the experimental data is from Ref. [13]. The thin lines are
calculations with fixed nuclei and the thick lines are the cor- calculations with fixed nuclei and the thick lines are the cor-
respondingcalculations includingvibrations. respondingcalculations includingvibrations.
clear motion are significantly closer to the experimental we have considered how nuclear vibrations can be taken
data but the agreement is still poor. into account within the adiabatic theory and the single-
InFig.7weshowcalculationsandexperimentaldataat active electron approximation.
the wavelength of 1365nm, i.e., in the tunneling regime. All models did correctlypredict absence of suppressed
Theexperimentswereperformedatsuchahighintensity ionization for N :Ar and presence of suppressed ioniza-
2
that saturation effects become important in all our cal- tion for O :Xe. Quantitative comparisons were made
2
culations and correspondingly, the ratios are all around difficult due to disagreements between different experi-
unity. Thesepredictionsareinagreementwiththeexper- ments,andfurtherexperimentswouldthereforebehighly
iment for S2:Xe, Fig. 7 (a), andNO:Xe, Fig.7 (c). From desirable.
Fig.7(b)weseethattheSO:Xeratioismuchlowerthan
We find a rather good overall agreement between
unity. This is quite remarkable since all three ratios in
the MO-SFA LG theory and experiments on diatomic
Fig. 6 were nearly identical at 800nm but very different
molecules and their companion atoms, and we believe
at1365nm–suchwavelengthdependenciesareofcourse
that the MO-SFA LG model accounts quite well for
impossible to predict by quasistatic tunneling theories.
the structural dependency on the strong-field ionization
rates. In general we found the length gauge MO-SFA to
be in better agreementwith experiments than the veloc-
V. CONCLUSIONS ity gauge MO-SFA – in particular in the case of D :Ar
2
where the MO-SFA VG predicted no suppression of ion-
We have made a detailed study of strong-field ioniza- ization in contradication with the experimental findings
tion of diatomic molecules and their companion atoms. and the other theories.
In particular we have investigated whether the approxi- The MO-ADK predictions were mostly in accordance
mate models: MO-SFALG, MO-SFAVG andMO-ADK with previously published results using this model [3].
are able to correctly predict the presence or absence of For the cases N :Ar, D :Ar and O :Xe these predictions
2 2 2
suppressed ionization in certain molecules. Furthermore arealsoinqualitativeagreementwithexperimentaldata.
10
ThemostsignificantdifferencesbetweentheMO-ADK rect ratio between molecules and atoms with different
andthe MO-SFALGcalculationsarethe muchlowerra- ionization potentials. This indicates that the depen-
tios predicted by the latter theory in the cases O :Xe dency on the electronic binding energy is not correctly
2
andCO:Kr. InbothcasestheresultsoftheMO-SFALG accountedforandpointstotheneedforthedevelopment
model agreedbetter with the experiments. This is inter- of improved models, including electron-electron correla-
esting in view of recent experiments [40] which reported tions and exact final states.
suppressionofstrong-fieldionizationfortransitionmetal
atoms relative to expectations, but only compared with
tunneling theory, and not SFA. For the future work it
Acknowledgments
would be interesting to check the SFA LG model on the
transition metals to investigate if the suppression is due
toageneralbreakdownofthesingle-activeelectronmod- WewouldliketothankA.Saenzforusefuldiscussions.
els or if it is due only to a failure of the ADK model. L.B.M. is supported by the Danish Natural Science Re-
All the models considered here fail to predict the cor- search Council (Grant No. 21-03-0163).
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