Ab initio calculations are presented for the vibrational branching ratios in three-photon resonant four-photon ionization of H_2 via the C^1Π_u state. Energy and internuclear distance dependences of the boundfree electronic transition matrix element are explicitly included to estimate deviations from the Franck-Condon approximation. While our calculated branching ratios confirm certain important trends seen experimentally, some differences remain

Dixit, S. N.

Lynch, D. L.

McKoy, V.

English

Caltech Authors - Main

RAPID COMMlJNICA riONS 
PHYSICAL REVIEW A VOLUME 30, NUMBER 6 DECEMBER 1984 
Three-photon resonant four-photon ionization of H2 via the C 1ITu state 
S. N. Dixit, D. L. Lynch, and V. McKoy 
Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, 
Pasadena, California 91125 
(Received 4 September 1984) 
Ab initio calculations are presented for the vibrational branching ratios in three-photon resonant four-
photon ionization of H2 via the C 1ITu state. Energy and internuclear distance dependences of the bound-
free electronic transition .matrix element are explicitly included to estimate deviations from the Franck-
Condon approximation. While our calculated branching ratios confirm certain important trends seen ex-
perimentally, some differences remain. 
I. INTRODUCTION 
Resonant enhanced multiphoton ionization (REMPI) is a 
powerful probe of electronically excited states. The high-
power lasers used in such studies not only provide high 
selectivity in energy (of the order of the laser bandwidth), 
but also make accessible, via multiphoton absorption; states 
that are single-photon forbidden by dipole selection rules. 
When combined with photoelectron energy analysis, REMPI 
allows us to focus very directly on the photoionization of 
the excited states and on the dynamics of multiphoton ioni-
zation. Recent experiments1- 12 on REMPI processes in dia-
tomic molecules such as H2, CO, NO, and N2 have begun to 
reveal several interesting features. A theoretical under-
standing of these features is obviously needed. 
Recently, Pratt, Dehmer, and Dehmer12 analyzed the 
photoelectron energy spectrum resulting from a three-
photon resonant four-photon ionization of H2 via the C 1IIu 
state. This resonant state, Rydberg in character, overlaps in 
energy with the valence B 1Iu+ state. The rr+ component 
of this C 1IIu state interacts with the B 1I: state via the 
Rydberg-valence mixing, while the II- component is unaf-
fected. Pratt, Dehmer, and Dehmer12 were able to select 
only the rr- component of the C 1IIu state as the resonant 
intermediate state by tuning the laser frequency in reso-
nance with the three-photon Q ( 1) transition line. This il-
lustrates the tremendous selectivity achievable in REMPI 
processes. The kinetic energy of the photoelectrons ejected 
from this resonant intermediate state was analyzed. These 
data provide the branching ratios for ionization out of a 
specific vibrational level v' of the C 1IIu state into different 
vibrational level v + of the X 2I/ state of H 2+. These vi-
brational branching ratios deviated significantly from the ap-
propriate Franck-Condon factors. Possible reasons for this 
behavior are the energy and the internuclear distance 
dependence of the electronic transition moment, autoioniza-
tion, and perturbations with overlapping, allowed transi-
tions. Autoionization and perturbations are Jess probable 
due to other considerationsP In this Rapid Communica-
tion, we quantitatively assess the role of energy and R 
dependence of the electronic transition moment on the vi-
brational branching ratios in these. (3 + 1) REMPI experi-
ments of Pratt, Dehmer, and Dehmer,'2 i.e., 
3hv 
H2(X 1I/v".Jo= 1) __. Hnc 1Iluv',-"= 1) 
Comparison with the measured branching ratios shows good 
agreement for vibrational branching ratios for v' = 0 and 1 
excitations although differences exist between theory and 
experiment for v' = 2, 3, and 4 excitations. 
II. THEORY 
An analysis of REMPI cross sections and the related an-
gular distributions requires molecular parameters such as 
the transition moments and transition frequencies and their 
incorporation into the dynamics equations. These steps are 
nontrivial even for the simplest of diatomic molecules H2 
and, hence, it is not surprising that very few theoretical cal-
culations have been carried out for molecular REMPI 
processes.13• 14 Our framework for analyzing the dynamics 
has been presented elsewhere.15 For the process in Eq. (1), 
the probability for ejection of an electron in the direction 
( (}, cp) with respect to the light polarization vector (linearly 
polarized light was used in the experiment of Ref. 12 and is 
assumed here) obeys 
dP(O, cp) 
dt 
(2) 
In this equation, pt'1 is the density matrix element 
(J1M'I pI -"M) (all indices other than -" and M needed to 
describe a molecular state have been suppressed for clarity) 
and r M M I a generalized differential ionization rate. De-
l I 
tailed expressions for r MMI are given in Ref. 15, where it is 
I I 
also shown that in the absence of M mixing, ionization 
through each 111M) forms an independent channel. Thus, 
p 111 is proportional to 8MMI' and, therefore, for photoioniza-
1 I 
tion of the C 1IIu state excited from the ground state via a 
three-photon Q (1) transition (-" = 1), Eq. (2) reduces to 
dP(O,cp) 
dt = fupn + fooPoo+ f -t-lP-1-1 (3) 
r M M 1 can in turn be expanded in spherical harmonics15 as 
I I 
(4) 
where the coefficients 'Y LM·MM' depend on the bound-free 
' I I 
radial matrix elements and scattering phase shifts. M = 0 
3332 ©1984 The American Physical Society 
RAPID COMMLINICA TIONS 
THREE-PHOTON RESONANT FOUR-PHOTON ... 3333 
for M = M/ and r M1M1 and P (9, r/J) in Eqs. (3) and (4), are independent of r/J and consist of only Legendre polynomials. 
Furthermore, as the accessible partial waves (u11 , 1r1 , and B1 ) are all of the same g symmetry only even order Legendre po-
lynomials up to L~~! = 4 [see Eq. (40) of Ref. 15] survive in Eq. (4). Thus, 
rM.M.= I 'Yw;MM Yw(9,r/J) . 
1 1 L=0,2,4 I I 
(5) 
Equation (3) shows that, in addition to r M,M1, p 11 ( i = 0, ± 1) are needed to calculate the actual REMPI angular distribu-
tions. For the weak-field excitation conditions of Ref. 12, p 11 is given by 
Pit oc I I 
Mo IJtMt> 
(JtMil;.t · €IJ2M2} (hM2I.U · E'lhMt) (JtMd;.t · E'IJoMo) 
(£,1 - E10 - hv) (£12 - £10 - 2hv) 
(6) 
IJ2M2} 
From the X 1I,+ initial state, the dipole allowed inter-
mediate states, at the one-photon level, are of I;i and Ilu 
symmetry, while those at the two-photon level are of I/, 
TI11 , and /:1 11 symmetry. For an overall three-photon Q ( 1) 
transition, J 0 =J1=1, which restricts J2 to be 1 or 2. Note 
that the two-photon transition from the Jo= 1 level of the 
X 1 I 11+ state to the J 2 = 0 level of a I/ -type state is forbid-
den by parity selection rules. Moreover, the h = 0 level 
does not exist in TI11- and /:111-type states. Of these allowed 
states, the J 2 = 2 state contributes only if the electronic state 
is of 1111 or 1:1, type. In H2, the lowest states of ll11 and 1:111 
symmetry are, respectively, the / 1116 and J 11:1 6 , which lie at 
about 14-15 eV above the ground state.l6 The E, F 1I 11+ 
state, on the other hand, is about 12 eV from the ground 
state. For photon energies hv- 4 eV, the detuning 
(E12 - E10 - 2hv) for these 116 and 1:1 6 states is about 6 eV 
while that from the E, F 1 I/ state is about 4 e V. Hence, in 
the preliminary results presented here, we neglect the con-
tribution to Pu from states other than those of I 6+ sym-
metry. Such contributions will be included in later studies. 
Neglecting this contribution eliminates the need to calculate 
any p 11 , if one is interested only in the relative branching ra-
tios. This comes about as p00 =0 for I,+ states with h= 1. 
Furthermore, Pll = P-t-I> and ru =f -t-t which, com-
bined with Eq. (3) imply 
(7) 
In the absence of saturation as assumed here, the solution 
of the above equation is 
P(9)=constxr11 (9). (8) 
Since we are only interested in relative contributions, we 
shall set the constant in the above equation to unity and 
use, hereafter, 
P(9)=f 11 (9) . (9) 
In the same spirit, we also ignore factors such as the laser 
intensity in the expression for r 11 (9) [see Eq. (29) of Ref. 
15]. 
Thus, to investigate relative branching ratios, all we need 
to calculate is the differential cross section for ionization out 
of the I J1 = l,M1 = l} state. From the analysis in Ref. 15, 
r 11 (9) depends on the bound-free electronic transition ma-
trix elements of the type 
(10) 
where 
(11) 
denotes the transition moment for the i- f transition at a 
given internuclear separation R and electron momentum k. 
In our studies of the non-Franck-Condon effects on vibra-
tional branching ratios, we adopt the following notation. (a) 
Franck-Condon (FC): r)/">(k,R) inside the integral in Eq. 
(10) is replaced by its value at some k2 = k~ and R = Re. 
rJt> th(m simplifies to a product of r)/"l(k0 ;Re) and the 
Franck-Condon overlap between the v+ vibrational level of 
the ion and v' vibrational level of the C 1IIu state. (b) 
Non-Franck-Condon (non-FC): rjr>(k;R) in Eq. (10) is 
replaced by rj,"'>(k0 ;R) and is retained inside the R integral. 
These calculations assess the effect of the R dependence of 
the transition matrix element on the branchin~ ratios. (c) 
Full: here both the k and R dependences of rfi~'l(k;R) are 
retained. For a fixed photon energy hv, selecting v' and v+ 
fixes k 2 by the energy conservation equation (in atomic 
units): 
k2 
Ev,+hv=Ev++T (12) 
These results include both the energy dependence as well as 
the R dependence of the cross sections and should, there-
fore, be most complete. 
In our calculations, the C 1IIu wave function was obtained 
using the improved-virtual-orbital (IVO) techniqueP The 
energies of the C 1IIu state obtained this way as a function 
of R are within 5% of the correct values. 16 The continuum 
wave functions for the photoelectron in u 11, 1r 11 , and 81 
channels were calculated by solving the Hartree-Fock equa-
tions using the iterative Schwinger variational techniques. 18 
rj,"'>(k;R) was then calculated at R = 1, 1.4, 2, 3, and 5 a.u. 
and for a range of k (0.05-0.45 a.u.). Interrolation was 
performed in R and in k to obtain required r)/" ( k;R). For 
part (a), Re= 1.4 a.u. The value of k~ for parts (a) and (b) 
was determined from Eq. (12) with v' = v + and hv taken 
from Ref. 12 for excitation to the Q ( 1) branch of 
C 1IIu( v') state. For part (c), the correct value of k is 
determined from Eq. (12) for each v +. Finally, the vibra-
tional wave functions Xv' and Xv+ were calculated using the 
finite element method of Malik, Eccles, and Secrest19 with 
the potential curves of Sharp. 16 · 
RAPID COMMlJN ICA I IO!'.'S 
3334 S. N. DIXIT, D. L. LYNCH, AND V. McKOY 
III. RESULTS 
In Fig. 1 we compare the branching ratios calculated at 
the various levels of approximation (a), (b), and (c) with 
the experimental results of Pratt et a/. 12 As in the experi-
mental data, we plot P(8 = 0). The results are normalized 
such that the v 1 = v + peak in all three approximations and 
in the experiment is of unit height. (Note the break in the 
graph for the V1 =v+ peak.) As expected, the vl=v+ peak 
is dominant. This is due to the Rydberg character of the 
C 11Iu state which makes the potential surface for the C 11Iu 
state nearly identical to (but shifted in energy from) that of 
the X 2Ig+ state of Ht. The theoretical branching ratios de-
crease rapidly for v+ ~ V 1• The results of approximations 
(a), (b), and (c) reveal an interesting feature: the branch-
ing ratios decrease for v+ < V 1 as the R and k dependences 
of r}t> are included and increase for v+ > V 1 with these 
dependences included. The difference between "non-
Franck-Condon" and "full" results is simply a reflection of 
1.0 1 
0.3 1-
0.2 5 
0.2 
0.15 
0.1 
0.0 5_ 
-
1.0 
0.0 8_ 
-
0.0 6_ 
-
0.04 
-
-
0.0 2-
0 
1.0 
0.04 
f-
r-
:0: 
1-
1-
1-
0.0 •2t-
f-
1 v' = 2 
n. ~ ~ 
.II. v'= 1 
I .J ' n.... 
v' =O 
.. 
~·· 
4 3 0 
the increasing of the cross section for decreasing energy 
which skews the branching ratios towards higher v + values. 
[Note that, since v+ is increasing to the left, k increases to 
the right as indicated by Eq. (12).] The difference between 
Franck-Condon and non-Franck-Condon results arises from 
the particular R dependence of the photoionization cross 
section for the C 11Iu state. This difference is probably 
specific for the photoionization process under study and 
may be different for other states and molecules. 
We now compare the results of our studies with the data 
of Pratt et a/. 12 For v 1 = 0 and v 1 = 1 excitations, the agree-
ment between theory and experiment seems quite good. 
For v 1 = 2-4, the experimental branching ratios for v + ~ v' 
are much larger than the theoretical predictions. In particu-
lar, for v' = 4, the v+ = 3, 5, and 6, experimental peaks are 
all of about equal height, while theory predicts the v + = 3 
and 6 peaks to have 0.11 and 0.2 times the height of v + = 5 
peak (which is -18% of v+=4 peak). A similar 
discrepancy exists in the v' = 3 and v 1 = 2 excitation data as 
1.0 £ 
0.3 r-
0.25 1-
0.2 1-
0.15 f-
0.1 
0.0 5 
1.0 1-
0.35 
1- • 
0 
0.3 1- II 
r::.il 
0.25 f-
0.2 1-
0.15 f-
0.1 1-
0.0 f-
rt I 
EXPT 
FULL 
NON-FC 
FC 
l IlL 
6 5 
I 
I 
4 
v+ 
v'=4 
~ 
1 v' = 3 
~ 
3 2 
FIG. 1. Vibrational branching ratios in (3 + 1) REMPI of H2 via the C 1 llu state. v' denotes the vibrational state of C 1nu state and v + 
that of the X 2Ii' state of the ion. 
RAPID COMMUNICATIONS 
THREE-PHOTON RESONANT FOUR-PHOTON ... 3335 
well. These differences may be due to autoionization, ac-
cidental resonances at the three-photon excitation level, and 
the contribution of the Il8 and l:i8 states to the three-photon 
excitation amplitude, which are neglected in our present 
studies. Of these, autoionization seems to play no role as 
the results of Ref. 12 are insensitive to changes in photon 
frequency. To analyze the contribution of Il8 ami l:i 8 states 
to the excitation, we have calculated r 00 in Eq. (3) and ob-
serve that the branching ratios in f 00(9=0) are very similar 
to those quoted here. Thus, the inclusion of the r ooPoo 
term in Eq. (3) will not significantly alter the relative 
branching ratios. The existence of accidental resonances 
with the P branch of some high vibrational level of the 
B 1Iu+ state can also be ruled out for all v' states except 
v' = 3 (Ref. 12). Another improvement in our calculations 
would be the inclusion of correlation effects. Such studies 
are currently underway. 
In summary, we have presented ab initio calculations for 
vibrational branching ratios in (3 + 1) REMPI of H2 via the 
C 1Ilu state. Calculated ratios are in good agreement with 
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recent experimental results12 for excitation through lower 
vibrational level of the C 1Ilu state. Differences do exist for 
excitation through higher vibrational levels. Further studies 
are needed to understand these differences. 
ACKNOWLEDGMENTS 
The authors wish to thank R. J. Cave and V. K. Babamov 
for providing us with the finite element program used for 
calculating the vibrational wave functions. This research 
was supported by the National Science Foundation under 
Grant No. CHE-8218166. The research reported in this pa-
per made use of the Dreyfus-National Science Foundation 
Theoretical Chemistry Computer which was funded through 
grants from the Camille & Henry Dreyfus Foundation, the 
National Science Foundation (Grant No. CHE-7820235), 
and the Sloan Fund of the California Institute of Technolo-
gy .. 
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Phys. 77, 877 (1982); see also S. N. Dixit and V. McKoy, ibid. 
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(1982); Phys. Rev. A 29, 183 (1984). 
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Three-photon resonant four-photon ionization of H_2 via the C^1 ∏_u state

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Three-photon resonant four-photon ionization of H_2 via the C^1 ∏_u state

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