Computational Studies of Molecular Frame Photoelectron
Angular Distributions
Robert R. Lucchese
Department of Chemistry
Texas A&M University
College Station, TX 77843-3255, U. S. A.
Abstract. Recent advances in molecular photoionization studies have allowed for the complete experimental
determination of all relevant experimental observables in selected cases. The quantitative understanding of this data
presents significant challenges for the theoretician. We will examine recent nearly complete experiments that use
coincidence measurements that detect both the photoelectron and the ionic photofragment from a given photoionization
event to determine the molecular frame photoelectron angular distributions (MFPAD). In favorable cases, from the
MFPAD one can obtain the underlying photoionization matrix elements. We will consider the level of agreement
between theory and experiment obtained with current computational methods for valence ionization in NO. We will
show that an extensive treatment of correlation is needed in order to obtain good agreement between theory and
experiment.
Nearly complete experiments of photoionization
have also been performed using rotationally resolved
two-photon ionization [3,4]. These experiments are
complementary to the experiments based on
photodissociation since they are based on rotationally
resolved final states, whereas the photodissociation
experiments usually involve short lived ion states
where it is not be possible to resolve the rotational
final states. A further limitation of the two-photon
experiment is that the ionization process that is being
probed is that of an excited state which may be of
limited interest.
INTRODUCTION
Recent experimental advances in the study of
molecular photoionization has allowed for a
qualitatively new level of detail to be probed
experimentally.
These experiments have studied
dissociative ionization processes and have measured
the velocities of the photoelectron and ionic fragments
in coincidence. When a diatomic molecule dissociates
in a time that is much shorter than the rotational
period, the direction of the motion of the fragment ion
can be assumed to give the molecular orientation at the
moment of the initial photoionization event.
Assuming a short dissociative lifetime is referred to as
the axial recoil approximation. When the axial recoil
approximation is valid, the coincidence experiment
gives a direct measurement of the molecular frame
photoelectron angular distribution (MFPAD). These
measured MFPADs have then been used to deduce the
underlying photoionization matrix elements [1,2]. In
specific cases it is then possible to perform a complete
experiment in the sense that all underlying matrix
elements that determine the behavior of the
photoionization process can be obtained.
The detailed information provided by these
experiments provides a very exacting test of the
theoretical description of the photoionization process.
For small linear molecules it is possible to perform
accurate calculations of the matrix elements that are
determined from the experiments. These calculations
can include both initial state and final state correlation.
In the final ionized state the correlation effects can be
classified as being either dynamic correlation in which
the electrons of the target respond to the motion of the
photoelectron and static correlation which is the
correlation of the motion among the target electrons.
In this paper we will review recent studies that have
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
791
photoionization by linearly polarized light with
polarization in the nˆ direction leading to the q final
state is then given by
compared computed and experimental dipole matrix
elements and examine the extent to which the theory is
able to represent the experiment.
I (θ k , φ k ,θ n , φ n ) =
THEORY AND COMPUTATIONAL
METHODS
Nc
M
(
Nc N b
)
(
p=1µ =1
p=1
)
∑ ∑ Iq,lmµ Iq,* l′,m′,µ′
M i ,M f l,m,µ
L ′,M l ′,m ′,µ′
L=0,2
m+µ′
×(−1)
The method we use to compute photoionization
matrix elements and the subsequent cross sections is
the multichannel Schwinger configuration interaction
(MCSCI) method [5,6].
In this approach the
photoionized state of a linear molecule for the
photoelectron leaving with momentum k in ion state q
with target electron angular momentum about the
molecular axis of Mf is represented as an expansion in
ion states of the form
Ψq,kf = ∑ A Φ pφ p = ∑ ∑ C pµ A ψ µφ p ,
4π 2 E
cgi
 (2l + 1)(2l ′ + 1) 1 2


 (2L + 1)(2 L ′ + 1)
× l, l ′, 0,0 L ′,0 l, l ′, − m, m′ L ′, M (5)
× 1,1,0, 0 L, 0 1,1, −µ, µ ′ L, − M
( )
∗
×Y L ′M Ωkˆ Y LM
(Ωnˆ )
where gi is the degeneracy of the initial state and the
terms l, l ′, m, m′ L, M are the usual Clebsch-Gordan
coefficients.
(1)
If one defines the molecular frame coordinate
system such that the molecule is oriented along the z
axis and the polarization direction is in the xz plane so
the φn = 0, then the molecular frame angular
distribution can be written as a sum of four functions
of the polar angle θk, which is the angle between the
direction of the emitted photoelectron and the
molecular axis, as follows [2]:
where Φp is the pth ion state, φp(r) is the corresponding
wave function for the photoelectron. The state Φp is a
configuration interaction (CI) wave function
constructed from Nb configuration state functions, ψµ.
The operator A is the antisymmetrization operator. It
is possible to explicitly indicate the dependence of the
M
function Ψq,kf on the direction of the photoelectron k
I (θ k , φ k ,θ n ) = F00 (θ k ) + F20 (θ k )P20 (cos θ n )
by using a partial-wave expansion
12
2
M
Ψq,kf =  
π 
∑ i l Ψq,klmYlm∗ (Ωkˆ ),
Mf
+F21 (θ k )P21 (cosθ n )cos(φ k ) . (6)
+F22 (θ k )P22 (cosθ n )cos(2φ k )
(2)
l,m
The FLN functions can be in turn expanded in Legendre
polynomials
where Ωkˆ stands for the polar and azimuthal angles
(θ k , φ k ) .
The partial wave dipole matrix elements are
FLN (θ k ) = ∑ C L ′LN PLN′ (cos θ k ),
then:
(7)
L′
12
L
I q,lm
µ = (k )
M
f
ΨiM i rµ Ψq,klm
(3)
where
C L ′LN =
and
V
I q,lm
µ
12
k)
(
=
E
2πE
∑
cgi (1+ δ N ,0 ) M ,M
i
ΨiM i
Mf
∇ µ Ψq,klm
,
(4)
∑ Iq,lmµ Iq,* l′,m′,µ′
f
l,m,µ
l ′,m ′,µ′
 (2l +1)(2l ′ +1)(L − N )!(L ′ − N )!1 2
×

(L + N )!(L ′ + N )!


where the superscripts L and V stand for the length
and velocity forms of the cross section, the superscript
Mi is the angular momentum of the initial electronic
state about the molecular axis, E is the photon energy,
and the operators rµ and ∇µ are the length and velocity
operators [7]. The differential cross section for
m+µ′
×(−1)
(8)
l, l ′,0, 0 L ′,0 l, l ′,−m, m′ L ′, N
× 1,1,0,0 L,0 1,1,−µ, µ ′ L,−N
In favorable experimental situations it is possible to
determine experimentally the values of the Ilmµ from
792
[2]. In Fig. 1 we show the MFPADs for the ionization
of NO leading to the c 3Π of NO+ at photon energies of
25 eV, 30 eV, 35 eV, and 40 eV. In each case we give
the MFPAD where the direction of polarization of the
field is parallel to the molecular axis. In order to
illustrate the sensitivity of the computed results to the
dynamic correlation, we show the results of both a
one-channel calculation where only the c 3Π state of
NO+ is included in the calculation and the results of
our largest calculation that contained 17 ion states.
We have also included in this figure the MFPADs of
the two experiments that have been performed on this
system. For both of the experimental distributions we
have taken the fitted dipole matrix elements and
generated the corresponding MFPAD using Eqs. (6)(8). There are two main features of the MFPADs
shown in Fig. 1. First one can see, particularly in the
17 channel calculation, the presence of the strong fwave scattering in the 30 eV results. This is the region
of the shape resonance in this channel that
qualitatively corresponds to the σ* state in NO which
would be expected to have the strong f-wave
characteristic seen in Fig. 1. The second feature of the
computed cross sections is that the MFPAD is most
sensitive to the inclusion of more correlation at low
energy with somewhat less sensitivity at higher
energy. One can also see that the inclusion of
interchannel coupling effects is important to obtaining
reasonable agreement between theory and experiment.
the measured values of the FLN. With linearly
polarized light it is not possible to determine the
overall sign of the phases of the matrix elements.
However, if the MFPAD is measured for circularly
polarized light this overall sign of the phases can be
determined [1]. The possibility that an experiment can
determine all of the unique value of the Ilmµ depends
on the symmetry of the initial and final states in the
system. By an analysis of the number of independent
variables needed to determine the Ilmµ compared to the
number of variables that can be determined from the
measured FLN one can make the following conclusions
[2]. For a Σ to Σ photoionization transition, the F00 and
F20 distributions contain enough information to
determine all the unique values of the Ilmµ except for
the phase between the parallel (µ = 0) and
perpendicular (µ = 1) transitions. If only one of the
initial and final target states is a Σ state, then a
complete experiment is performed if one measures the
F00, F20, F21, and F22. Finally, if neither the initial nor
the final target states are Σ states, measuring the four
FLN functions does not provide in general enough
parameters to determine the Ilmµ elements.
VALENCE IONIZATION OF NO
Consider the valence ionization of NO. On of the
strongest valence dissociative ionization channels is
the ionization leading to the c 3Π state of NO+. This
state rapidly dissociates into N+ (3P) and O (3P) atoms.
The c 3Π state can be qualitatively described as a (4σ)–
1
hole state. Experimental MFPADs have been
measured for this process at 23.64 eV [2] and 40.8 eV
[1]. In both cases, dipole matrix elements have been
deduced from the measured MFPADs. This process is
of the type discussed above, a Π to Π process, that in
principle cannot be used to obtain the dipole matrix
elements. However, from computational results, we
know that the matrix elements can be well
approximated by assuming that they have the same
symmetry as the matrix elements of a Σ to Σ process,
thus allowing for a determination of the Ilmµ from the
measured FLN functions.
OTHER SYSTEMS
In addition to NO, other systems for which we have
computed MFPADs include the 1s ionization of N2
[8], and the valence ionization of O2 [9] and N2O [10].
In the study of the 1s ionization of N2 [8] there is an
interesting feature that the two 1s hole states are nearly
degenerate. In some calculations where both of these
photoionization channels are included, strong coupling
between these two channels has been seen. This is
especially true in the region of the σ* shape resonance
which is only directly accessible when the hole is
formed in the 1σg orbital. However, we found that the
strength of this interaction can be overestimated if
correlated ion states are not used. Very good
agreement with experimental results [11] for the
MFPAD for the parallel transition was also found for
this system.
We have previously theoretically studied this
photoionization process [5].
Using the matrix
elements from the previous study we computed
MFPADs that were found to be in only fair agreement
with the experimental measurements. These earlier
calculations included only five channels in the
expansion given in Eq. (1).
We subsequently
performed calculations which include 17 channels that
lead to much better agreement with the experiments
In the study of the MFPAD of O2 [9] for ionization
leading to the B 2Σg– state of O2+, we found that good
agreement could only be obtained when the effects of
793
25 eV
30 eV
35 eV
40 eV
One
Channel
Seventeen
Channel
Exp.
FIGURE 1. MFPADs for ionization of NO leading to the c 3Π state of NO+. The one channel and seventeen channel results are
MCSCI results. The experimental result listed under the 25 eV photon energy is a 23.64 eV experiment from Ref. [2] and the
experimental result listed under 40 eV is a 40.8 eV experiment from Ref. [1]
the rotation of the molecule were included in the
calculation. This is an example where the axial recoil
approximation is not valid yet there is still information
about the MFPAD due to the fact that the rotational
states of the excited ion are polarized in the initial
ionization event.
CONCLUSION
A brief review of recent computed MFPADs for
linear molecules has been given. By considering the
example of the ionization of NO leading to the c 3Π
state we have seen that extensive dynamic correlation
must be included in the calculation in order to obtain
agreement with measured experimental MFPADs. We
have further seen that these dynamic correlation
effects are more important near threshold than at high
energy.
In the study of N2O [10] we found that for
ionization leading to the C 2Σ+ state of N2O+ there was
again a significant lifetime of the ion state leading to a
breakdown in the axial recoil approximation.
However, by comparing the computed and measured
MFPADs as a function of an assumed lifetime for the
C 2Σ+ state, we could estimate the lifetime as being
approximately 2 ps.
794
ACKNOWLEDGMENTS
The support of the National Science Foundation
(USA) through Grant No. INT-0089831 is gratefully
acknowledged. This work was also in part supported
by the Welch Foundation under grant No. A-1020 and
by the Texas A&M University Supercomputing
Facility.
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