In this paper, we report the results of ab initio calculations of photoelectron angular distributions and vibrational branching ratios for the (2+1) REMPI of H_2 via the E, F^(1)Σ^+_g  state, and compare these with the experimental data of Anderson et al. [Chem. Phys. Lett. 105, 22 (1984)]. These results show that the observed non‐Franck–Condon behavior is predominantly due to the R dependence of the transition matrix elements, and to a lesser degree to the energy dependence. This work presents the first molecular REMPI study employing a correlated wave function to describe the Rydberg–valence mixing in the resonant intermediate state

Dixit, S. N.

Huo, Winifred M.

Lynch, D. L.

McKoy, V.

Rudolph, H.

English

Caltech Authors - Main

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{2+1) resonant enhanced multiphoton ionization of H2 via the E,F 1l:t state 
H. Rudolph, D. L. Lynch,a> S. N. Dixit, and V. McKoy 
Arthur Amos Noyes Laboratory of Chemical Physics,h> California Institute of Technology. 
Pasadena, California 91125 
Winifred M. Huoc> 
Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 
(Received 30 September 1986; accepted 7 November 1986) 
In this paper, we report the results of ab initio calculations of photoelectron angular 
distributions and vibrational branching ratios for the (2 + 1) REMPI of H2 via the E,F 1 ~t 
state, and compare these with the experimental data of Anderson et a!. [Chern. Phys. Lett. 
105, 22 ( 1984)]. These results show that the observed non-Franck-Condon behavior is 
predominantly due to the R dependence of the transition matrix elements, and to a lesser 
degree to the energy dependence. This work presents the first molecular REMPI study 
employing a correlated wave function to describe the Rydberg-valence mixing in the resonant 
intermediate state. 
INTRODUCTION 
Resonant enhanced multiphoton ionization (REMPI) 
techniques provide a useful tool for the study of highly excit-
ed states and of the dynamics of the photoionization of these 
rovibrationally selected states. Furthermore, for systems 
with inversion symmetry, REMPI studies can be effectively 
used to access dipole-forbidden bound and continuum 
states. Several groups have experimentally exploited these 
advantages to demonstrate interesting aspects of REMPI 
dynamics. '-9 In a previous paper, 10 we have discussed the 
development of our theoretical framework for analyzing 
REMPI processes in diatomic molecules. In this theory, an 
(n + m) REMPI is viewed as a two-step process with n-
photon absorption to the resonant intermediate state fol-
lowed by an m-photon ionization out of it. Coupling the 
formalism with ab initio calculations of molecular proper-
ties, we have analyzed several recent REMPI experiments 
with encouraging results. 11- 14 
Recently Zare and co-workers15 have measured the 
photoelectron angular distributions and vibrational branch-
ing ratios in a one-color ( 2 + 1 ) REMPI of H 2 via the 
E,F 1 ~t state: 
2hv 
H 2 (X 1 ~t ,v0 ,J0 ) -+ Ht(E,F '~t ,v0 J;) 
hv 
-+ H 2+ (X 2 ~t ,v+) +e-. (1) 
The low resolution ( 150 me V) photoelectron spectra dis-
play highly non-Franck-Condon behavior as seen by the de-
parture of the vibrational branching ratios from Franck-
Condon values and by a strong vibrational state dependence 
of the photoelectron angular distributions. The potential en-
ergy curve of the E,F 1 ~g+ state has the well-known double-
well structure 16' 17 with the nature of the electronic wave 
•> Present address: Theoretical Division, Los Alamos National Laboratory, 
Los Alamos, New Mexico 87545. 
b> Contribution No. 7490. 
c> Mailing address: NASA-Ames Research Center, MS230-3, Moffett 
Field, California 94035. 
function changing from Rydberg-like in the inner well (E 
state) to valence-like in the outer well (F state). This pre-
sents an interesting situation for theoretical investigation as 
the electronic wave function for the E,F state has to be de-
scribed in a multiconfigurational scheme to incorporate the 
continuous change of its character with internuclear dis-
tance (R). We have, therefore, carried out ab initio calcula-
tions of photoelectron angular distributions and vibrational 
branching ratios for the REMPI process of Eq. ( 1). Our 
results are in general agreement with the measured values 15 
and indicate that the observed non-Franck-Condon behav-
ior results primarily from the R dependence of the electronic 
transition moments. In contrast to our earlier REMPI stud-
ies in which the resonant state was described by a single 
electronic configuration, i.e., the improved virtual orbital 
(IVO) approximation, 18 we have used a correlated wave 
function to described the Rydberg-valence mixing in the E,F 
state. Rydberg-valence mixing is a common feature in the 
spectrally congested region and must influence REMPI pro-
cesses through the highly excited states. 5_9 As such, the pres-
ent study, due to the isolated and "clean" nature of the E,F 
state, forms a prelude to forthcoming investigations of such 
effects in heavier diatomic molecules. An additional motiva-
tion for investigation of the REMPI process ofEq. ( 1) is the 
proposed use of this REMPI scheme for detecting product 
H 2 molecules in reactions. 19 
In the following we briefly review our theoretical ap-
proach and present the relevant details of the calculations. In 
the last section we discuss the calculated vibrational branch-
ing ratios and photoelectron angular distributions and com-
pare them to the experimental results of Anderson et al. 15 
THEORY 
The general theory for REMPI molecular processes is 
given in Ref. 10, and here we briefly describe only those 
details which are necessary for the specific scheme given in 
Eq. ( 1 ). As in the experiments, we assume that the mole-
cules are excited from an isotropic ground state by linearly 
polarized light of weak intensity. Under these conditions the 
1748 J. Chem. Phys. 86 (4), 15 February 1987 0021-9606/87/0417 48-04$02.1 0 © 1987 American Institute of Physics 
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Rudolph eta!. : Multiphoton ionization of H, 1749 
population Pu of each IJ;M;) level of the resonant interme-
diate state is proportional to10 
"''I~J,M,> (J;M;IweiJ1M1)(J1M11weiJoMo) 12 p .. cx:£... 11 
Mo E 1 - E0 - hv ' (2) 
where IJoM0) is the ground X 1~t state with angular mo-
mentum J0 and its z projection M0 , IJ1M1) is a virtual state 
connected to the ground state by a one-photon transition and 
hv is the photon energy. In the absence of M-mixing interac-
tion, e.g., collisions, each IJ;M;) level will form an indepen-
dent ionization channel, and the probability P( e,¢) of eject-
ing an electron in the direction ( e,¢) with respect to the 
polarization vector (e) of the light is given by10 
dP~,</J) = ~ r M,,M, ( {},</J )p ;; , (3) 
where r M,,M, ( e,¢) is the ionization width defined by Eq. 
( 29) of Ref. 10. P( (},¢) itself can be expanded in Legendre 
polynomials PL (cos{}) as 
P({}) = 2: f3 LPL (cos{}). (4) 
L=O 
For these weak field studies Lmax is determined by the rota-
tional quantum state of the intermediate level and is, for the 
( 2 + 1) REMPI process ofEq. ( 1), the smaller of2J; + 2 or 
6. The infinite summation in Eq. (2) goes over the complete 
set ofrovibrational electronic states IJ1M 1) ofH2. We have 
included in this summation only the lowest states of unger-
ade symmetry (B 1 ~u+ ,C 1 llu ) and kept up to v = 9 in the 
vibrational manifolds at the one-photon level. The relative 
vibrational branching ratios for Q(O) excitation are unaf-
fected by this truncation. For the Q( 1) excitation, based on 
the study of Huo and Jaffe,20 we estimate the truncation 
error to be about 10% in the relative populations of the 
MJ = 0 and MJ = 1levels. The summation over rotational 
states was performed implicitly using closure relations. 21 
The bound-bound transition moments were taken from pre-
vious calculations by W olniewicz22 and W olniewicz and 
Dressler, 23 and the vibrational wave functions were obtained 
numerically using the potential curves compiled by Sharp. 17 
The electronic wave function for the E,F 1 ~t state was de-
termined from a configuration interaction ( CI) calculation 
using a (6s5pld) uncontracted Gaussian basis set. This cal-
culation produces 240 configurations of the proper overall 
1 ~g+ symmetry. The wave function was determined at the 
internuclear distances R (a.u.) = 1.00, 1.401, 1.911, 2.50, 
3.12, 3.70, 4.39, 5.00 which cover the relevant range of the 
potential. The resulting energies are within 0.6% of the ac-
curate calculated values. 17 The photoelectron continuum 
wave functions for the ku u and k1r u channels were calculat-
ed in the frozen-core Hartree-Fock approximation using the 
iterative Schwinger variational method.24 Finally, the 
bound-free electronic transition moments (X 1~g+ ;ku u, 
k1r u I WE IE,F 1 ~t ) were calculated for each R and for a 
range of photoelectron kinetic energies, k 2/2, between 1.22 
and4.12 eV. Interpolation was performed in bothR andk to 
obtain the necessary matrix elements for integration over the 
numerical vibrational wave functions and for specific kinetic 
energies. 
l.001D (A) [D (B) 
0.15 
.2 
'Cil 
.... 
Oil 
c 0.10 :.a 
(.) 0.20 
c 
"' .... ~ 
0.05 0.10 I 
0.00 
0 1 2 
I I 0.00 ld l 
V+ 0 1 
R ~~ 
2 3 4 5 1h 
1.001 (C) ID 
D EXP 
0.25 
.FULL 
.g 0.20 
"' .... 
12lN-FC 
10 FC 
Oil 
c 0.15 :.a 
(.) 
c 
"' .... 0.10 ~ 
0.05 
O.OO-'-'-r---L~.-.LIIIM....l..llflo.....u~~~...,___u..,'""-U..,......u~ 
o 1 2 3 4 5 6 7 8 9 v+ 
FIG. I. Vibrational branching ratios for (2 + 1) REMPI via (A) vE F = 0, 
Q(O) level; (B) vE,F = 3, Q(l) level; (C) vE.F = 6, Q(l) level. ' 
RESULTS 
Figure 1 shows our calculated vibrational branching ra-
tios for the v; = vE,F = 0,3,6levels of the E,Fstate. For con-
venience the plotted data are reproduced in tabular form in 
Table I. The vibrational wave functions for V; = 0,3 states 
TABLE I. Comparison of the experimental and the theoretical vibrational 
branching ratios in three different approximations, as explained in the text. 
Entries less than 10-3 are not listed. 
v, v+ Exp• Full N-FC FC 
0 0 1.000 1.000 1.000 1.000 
0 1 0.156 0.068 0.064 0.033 
0 2 0.007 0.007 0.003 
3 0 0.126 O.D18 0.020 O.o38 
3 1 1.000 1.000 1.000 1.000 
3 2 0.359 0.116 0.112 0.049 
3 3 0.176 0.023 0.021 0.010 
3 4 0.066 0.003 0.003 
3 5 0.001 0.001 
6 0 O.D18 
6 1 0.182 0.034 0.036 0.055 
6 2 1.000 1.000 1.000 1.000 
6 3 0.187 0.081 0.083 0.002 
6 4 0.253 0.057 0.050 0.060 
6 5 0.146 0.005 0.005 0.001 
6 6 0.111 0.002 0.002 0.002 
6 7 0.076 0.002 0.003 0.017 
6 8 0.086 
6 9 0.073 0.005 0.004 0.014 
"From Ref. 15. 
J. Chern. Phys., Vol. 86, No.4, 15 February 1987 
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1750 Rudolph eta!. : Multiphoton ionization of H, 
are primarily located in theE well and look like vE = 0,1 
wave functions for a Rydberg state. The wave function for 
v; = 6 has considerable spread over the F well but still looks 
somewhat like avE = 2 wave function. 15•20 The plotted val-
ues are proportional to the probability of photoelectron ejec-
tion along the direction of the polarization of the light, 
P( f3 = 0, ¢ = 0). For each vibrational level of the ion four 
bars are shown. The experimental data is taken from Ref. 15. 
The three bars labeled Franck-Condon, non-Franck-Con-
don, and full correspond to our calculated results using dif-
ferent levels of approximation, 11 i.e., ( 1) neglecting the ener-
gy and internuclear distance dependence of the bound-free 
matrix elements, ( 2) neglecting the energy dependence but 
retaining the internuclear distance dependence, and ( 3) in-
cluding both the internuclear distance and energy depen-
dence, respectively. The plots are normalized to the most 
intense peak determined experimentally, 15 which, as expect-
ed from the Rydberg character of the E state, is 
llv=v + - vE = 0, where vE is the vibrational quantum 
number defined for the inner-well levels. The similarity 
between the non-Franck-Condon and full branching ratios 
indicates the weak dependence of the results on the kinetic 
energy of the photoelectron. However, the larger difference 
between the Franck-Condon and non-Franck-Condon re-
sults indicate a greater sensitivity to the internuclear dis-
tance dependence of the photoionization matrix elements. 
This points to the inadequacy of the Franck-Condon ap-
proximation. 15 The disagreement between the calculated 
and measured vibrational branching ratios are substantial, 
although it is difficult to be very specific at this time, the 
disagreement could arise, on the theoretical side, from our 
neglect of autoionization and saturation effects. On the other 
hand they may also be due to inherent difficulties in the 
experimental determination of such branching ratios. 
We compare the calculated and experimental photo-
electron angular distributions for the Q(O), and Q( 1) 
branches via the vE,F = 3 vibrational levels in Figs. 2 and 3, 
respectively. In these figures the vertical direction corre-
sponds to f3 = o•. The non-Franck-Condon behavior for 
v = 0 
/ 
' 
' ' ,
' 
' 
,
' ' 
, 
' ' 
' ' 
' ' 
' ' , __ .... / 
A B A 
V/= 2 
A B A 
V+= 1 
( \ 
I i 
I ( 
' 
/ 
;- ~ 
/ 
' ( \ 
i '! 
\ ( 
B 
--~ V+= 3 
/ ' ( \ 
I I 
I I 
I I 
\ / 
' / >< 
/ ' / \ 
I \ 
I I 
I I 
I I 
\ / 
'-~ 
B 
FIG. 2. Photoelectron angular distributions for the Q(O) branch via the 
vE,F = 3level. (A) Experimental results of Ref. 15, and (B) present full 
results. (J = o· is vertical. 
/---~~) \ 
I I 
I, / 
/ 
( 
I 
\ 
A 
A 
v = 0 
B 
v = 2 
B 
' 
' 
' 
' 
' 
FIG. 3. Same as Fig. 4 but via the Q( I) branch. 
A 
A 
B 
' / \ 
I I 
I I 
I I 
I I 
\ / 
' / >-, 
/ ' / \ 
I \ 
I I 
I I 
\ I 
"------/ 
B 
these transitions is seen by the dependence of the angular 
distributions on the final vibronic levels particularly for 
v + = 0 to v + = 1. Recent (3 + 1) REMPI studies in H 2 via 
the C 1TI" state have also demonstrated such non-Franck-
Condon effects on the photoelectron angular distribution. 26 
The calculated distributions reported are the "full" results 
where both the internuclear and photoelectron kinetic ener-
gy dependence are retained in the evaluation of the transition 
moments. A quantitative comparison between the calculat-
ed and measured angular distributions is difficult for reasons 
discussed in Ref. 15; however, the general trends are repro-
duced. The lack of "sharpness" around 90• for some of the 
experimental distributions may be attributed to the finite 
angular resolution (;::::: 3•) of the photoelectron detector. 
It is clear from the Figs. 2 and 3 that the angular distri-
butions for the Q(O) and Q( I) excitations are almost identi-
cal. The/14 ofEq. ( 4) in the latter case is about 5%-IO% of 
the /12 coefficient. This implies that either the alignment in 
the Q( I) excitation is small or, that the partial waves larger 
than the p wave are contributing weakly to the photoioniza-
tion. Indeed, our calculations confirm that both these effects 
contribute significantly to the REMPI, resulting in a sup-
pression of /14 . 
CONCLUSION 
We have presented results for the photoelectron angular 
distributions and vibrational branching ratios resulting from 
a (2 + I) one-color REMPI ofH2 via the vE = 0,3,6 vibra-
tional levels of the E,F 1~/ state. A correlated electronic 
wave function has been used to consistently describe the 
Rydberg-valence mixing in the E,F 1 ~/ state. The calculat-
ed vibrational branching ratios are in general agreement 
with the experimental results of Anderson eta/. 15 indicating 
that the pronounced non-Franck-Condon behavior ob-
served is primarily due to the internuclear distance depen-
dence of the electronic transition moments. General agree-
ment between the theoretical and experimental angular 
distributions is obtained. We have, in the present work, ne-
glected the effects of saturation and the possible presence of 
autoionization features in the continuum. In a previous pa-
J. Chem. Phys., Vol. 86, No. 4, 15 February 1987 
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Rudolph eta/. : Multiphoton ionization of H, 1751 
per14 we have shown the sensitivity of the calculated results 
to the inclusion of saturation effects. The effects of the Ryd-
berg-valence mixing on the photoelectron angular distribu-
tions and vibrational branching ratios are expected to be 
even more dramatic if the outer-well vibrational states 
vF = 0, 1,2,3, i.e., vE,F = v; = 1,2,4,5 were accessed. 
Further theoretical and experimental studies for ionization 
through the outer well would therefore be very useful in un-
raveling the dynamics of these molecular photoionization 
processes. 
ACKNOWLEDGMENTS 
This research was supported by the National Science 
Foundation under Grant No. CHE8521391. W. M. H. ac-
knowledges support from the NASA Ames Cooperative 
Agreell).ent NCC 2-147, and H. R. gratefully acknowledges 
support from the Danish Natural Science Research Council 
and the Carlsberg Foundation. 
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J. Chem. Phys., Vol. 86, No.4, 15 February 1987 


(2+1) resonant enhanced multiphoton ionization of H_2 via the E, F^(1)Σ^+_g  state

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(2+1) resonant enhanced multiphoton ionization of H_2 via the E, F^(1)Σ^+_g state

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